cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A171368 Another version of A126216.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 5, 0, 9, 0, 1, 0, 0, 0, 21, 0, 14, 0, 1, 0, 0, 14, 0, 56, 0, 20, 0, 1, 0, 0, 0, 84, 0, 120, 0, 27, 0, 1, 0, 0, 42, 0, 300, 0, 225, 0, 35, 0, 1, 0, 0, 0, 330, 0, 825, 0, 385, 0, 44, 0, 1, 0, 0
Offset: 0

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Author

Philippe Deléham, Dec 06 2009

Keywords

Comments

Expansion of the first column of the triangle T_(0,x), T_(x,y) defined in A039599; T_(0,0)= A053121, T_(0,1)= A089942, T_(0,2)= A126093, T_(0,3)= A126970.
T(n,k) is the number of Riordan paths of length n with k horizontal steps. A Riordan path is a Motzkin path with no horizontal steps on the x-axis. - Emanuele Munarini, Oct 14 2023

Examples

			Triangle begins:
  1 ;
  0,0 ;
  1,0,0 ;
  0,1,0,0 ;
  2,0,1,0,0 ;
  0,5,0,1,0,0 ;
  5,0,9,0,1,0,0 ;
  ...

Crossrefs

Cf. A000108, A001003,

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A099323(n+1), A126120(n), A005043(n), A000957(n+1), A117641(n) for x = -1, 0, 1, 2, 3 respectively.

A001147 Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).

Original entry on oeis.org

1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, 13749310575, 316234143225, 7905853580625, 213458046676875, 6190283353629375, 191898783962510625, 6332659870762850625, 221643095476699771875, 8200794532637891559375, 319830986772877770815625
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The solution to Schröder's third problem.
Number of fixed-point-free involutions in symmetric group S_{2n} (cf. A000085).
a(n-2) is the number of full Steiner topologies on n points with n-2 Steiner points. [corrected by Lyle Ramshaw, Jul 20 2022]
a(n) is also the number of perfect matchings in the complete graph K(2n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
Number of ways to choose n disjoint pairs of items from 2*n items. - Ron Zeno (rzeno(AT)hotmail.com), Feb 06 2002
Number of ways to choose n-1 disjoint pairs of items from 2*n-1 items (one item remains unpaired). - Bartosz Zoltak, Oct 16 2012
For n >= 1 a(n) is the number of permutations in the symmetric group S_(2n) whose cycle decomposition is a product of n disjoint transpositions. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001
a(n) is the number of distinct products of n+1 variables with commutative, nonassociative multiplication. - Andrew Walters (awalters3(AT)yahoo.com), Jan 17 2004. For example, a(3)=15 because the product of the four variables w, x, y and z can be constructed in exactly 15 ways, assuming commutativity but not associativity: 1. w(x(yz)) 2. w(y(xz)) 3. w(z(xy)) 4. x(w(yz)) 5. x(y(wz)) 6. x(z(wy)) 7. y(w(xz)) 8. y(x(wz)) 9. y(z(wx)) 10. z(w(xy)) 11. z(x(wy)) 12. z(y(wx)) 13. (wx)(yz) 14. (wy)(xz) 15. (wz)(xy).
a(n) = E(X^(2n)), where X is a standard normal random variable (i.e., X is normal with mean = 0, variance = 1). So for instance a(3) = E(X^6) = 15, etc. See Abramowitz and Stegun or Hoel, Port and Stone. - Jerome Coleman, Apr 06 2004
Second Eulerian transform of 1,1,1,1,1,1,... The second Eulerian transform transforms a sequence s to a sequence t by the formula t(n) = Sum_{k=0..n} E(n,k)s(k), where E(n,k) is a second-order Eulerian number (A008517). - Ross La Haye, Feb 13 2005
Integral representation as n-th moment of a positive function on the positive axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/2)/sqrt(2*Pi*x) dx, n >= 0. - Karol A. Penson, Oct 10 2005
a(n) is the number of binary total partitions of n+1 (each non-singleton block must be partitioned into exactly two blocks) or, equivalently, the number of unordered full binary trees with n+1 labeled leaves (Stanley, ex 5.2.6). - Mitch Harris, Aug 01 2006
a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is i for iDavid Callan, Sep 25 2006
a(n) is the number of increasing ordered rooted trees on n+1 vertices where "increasing" means the vertices are labeled 0,1,2,...,n so that each path from the root has increasing labels. Increasing unordered rooted trees are counted by the factorial numbers A000142. - David Callan, Oct 26 2006
Number of perfect multi Skolem-type sequences of order n. - Emeric Deutsch, Nov 24 2006
a(n) = total weight of all Dyck n-paths (A000108) when each path is weighted with the product of the heights of the terminal points of its upsteps. For example with n=3, the 5 Dyck 3-paths UUUDDD, UUDUDD, UUDDUD, UDUUDD, UDUDUD have weights 1*2*3=6, 1*2*2=4, 1*2*1=2, 1*1*2=2, 1*1*1=1 respectively and 6+4+2+2+1=15. Counting weights by height of last upstep yields A102625. - David Callan, Dec 29 2006
a(n) is the number of increasing ternary trees on n vertices. Increasing binary trees are counted by ordinary factorials (A000142) and increasing quaternary trees by triple factorials (A007559). - David Callan, Mar 30 2007
From Tom Copeland, Nov 13 2007, clarified in first and extended in second paragraph, Jun 12 2021: (Start)
a(n) has the e.g.f. (1-2x)^(-1/2) = 1 + x + 3*x^2/2! + ..., whose reciprocal is (1-2x)^(1/2) = 1 - x - x^2/2! - 3*x^3/3! - ... = b(0) - b(1)*x - b(2)*x^2/2! - ... with b(0) = 1 and b(n+1) = -a(n) otherwise. By the formalism of A133314, Sum_{k=0..n} binomial(n,k)*b(k)*a(n-k) = 0^n where 0^0 := 1. In this sense, the sequence a(n) is essentially self-inverse. See A132382 for an extension of this result. See A094638 for interpretations.
This sequence aerated has the e.g.f. e^(t^2/2) = 1 + t^2/2! + 3*t^4/4! + ... = c(0) + c(1)*t + c(2)*t^2/2! + ... and the reciprocal e^(-t^2/2); therefore, Sum_{k=0..n} cos(Pi k/2)*binomial(n,k)*c(k)*c(n-k) = 0^n; i.e., the aerated sequence is essentially self-inverse. Consequently, Sum_{k=0..n} (-1)^k*binomial(2n,2k)*a(k)*a(n-k) = 0^n. (End)
From Ross Drewe, Mar 16 2008: (Start)
This is also the number of ways of arranging the elements of n distinct pairs, assuming the order of elements is significant but the pairs are not distinguishable, i.e., arrangements which are the same after permutations of the labels are equivalent.
If this sequence and A000680 are denoted by a(n) and b(n) respectively, then a(n) = b(n)/n! where n! = the number of ways of permuting the pair labels.
For example, there are 90 ways of arranging the elements of 3 pairs [1 1], [2 2], [3 3] when the pairs are distinguishable: A = { [112233], [112323], ..., [332211] }.
By applying the 6 relabeling permutations to A, we can partition A into 90/6 = 15 subsets: B = { {[112233], [113322], [221133], [223311], [331122], [332211]}, {[112323], [113232], [221313], [223131], [331212], [332121]}, ....}
Each subset or equivalence class in B represents a unique pattern of pair relationships. For example, subset B1 above represents {3 disjoint pairs} and subset B2 represents {1 disjoint pair + 2 interleaved pairs}, with the order being significant (contrast A132101). (End)
A139541(n) = a(n) * a(2*n). - Reinhard Zumkeller, Apr 25 2008
a(n+1) = Sum_{j=0..n} A074060(n,j) * 2^j. - Tom Copeland, Sep 01 2008
From Emeric Deutsch, Jun 05 2009: (Start)
a(n) is the number of adjacent transpositions in all fixed-point-free involutions of {1,2,...,2n}. Example: a(2)=3 because in 2143=(12)(34), 3412=(13)(24), and 4321=(14)(23) we have 2 + 0 + 1 adjacent transpositions.
a(n) = Sum_{k>=0} k*A079267(n,k).
(End)
Hankel transform is A137592. - Paul Barry, Sep 18 2009
(1, 3, 15, 105, ...) = INVERT transform of A000698 starting (1, 2, 10, 74, ...). - Gary W. Adamson, Oct 21 2009
a(n) = (-1)^(n+1)*H(2*n,0), where H(n,x) is the probabilists' Hermite polynomial. The generating function for the probabilists' Hermite polynomials is as follows: exp(x*t-t^2/2) = Sum_{i>=0} H(i,x)*t^i/i!. - Leonid Bedratyuk, Oct 31 2009
The Hankel transform of a(n+1) is A168467. - Paul Barry, Dec 04 2009
Partial products of odd numbers. - Juri-Stepan Gerasimov, Oct 17 2010
See A094638 for connections to differential operators. - Tom Copeland, Sep 20 2011
a(n) is the number of subsets of {1,...,n^2} that contain exactly k elements from {1,...,k^2} for k=1,...,n. For example, a(3)=15 since there are 15 subsets of {1,2,...,9} that satisfy the conditions, namely, {1,2,5}, {1,2,6}, {1,2,7}, {1,2,8}, {1,2,9}, {1,3,5}, {1,3,6}, {1,3,7}, {1,3,8}, {1,3,9}, {1,4,5}, {1,4,6}, {1,4,7}, {1,4,8}, and {1,4,9}. - Dennis P. Walsh, Dec 02 2011
a(n) is the leading coefficient of the Bessel polynomial y_n(x) (cf. A001498). - Leonid Bedratyuk, Jun 01 2012
For n>0: a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j)^2 for 1 <= i,j <= n. - Enrique Pérez Herrero, Jan 14 2013
a(n) is also the numerator of the mean value from 0 to Pi/2 of sin(x)^(2n). - Jean-François Alcover, Jun 13 2013
a(n) is the size of the Brauer monoid on 2n points (see A227545). - James Mitchell, Jul 28 2013
For n>1: a(n) is the numerator of M(n)/M(1) where the numbers M(i) have the property that M(n+1)/M(n) ~ n-1/2 (for example, large Kendell-Mann numbers, see A000140 or A181609, as n --> infinity). - Mikhail Gaichenkov, Jan 14 2014
a(n) = the number of upper-triangular matrix representations required for the symbolic representation of a first order central moment of the multivariate normal distribution of dimension 2(n-1), i.e., E[X_1*X_2...*X_(2n-2)|mu=0, Sigma]. See vignette for symmoments R package on CRAN and Phillips reference below. - Kem Phillips, Aug 10 2014
For n>1: a(n) is the number of Feynman diagrams of order 2n (number of internal vertices) for the vacuum polarization with one charged loop only, in quantum electrodynamics. - Robert Coquereaux, Sep 15 2014
Aerated with intervening zeros (1,0,1,0,3,...) = a(n) (cf. A123023), the e.g.f. is e^(t^2/2), so this is the base for the Appell sequence A099174 with e.g.f. e^(t^2/2) e^(x*t) = exp(P(.,x),t) = unsigned A066325(x,t), the probabilist's (or normalized) Hermite polynomials. P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for A066325(x,t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), where UP(n,t) are the polynomials of A066325 and, e.g., (P(.,t))^n = P(n,t). - Tom Copeland, Nov 15 2014
a(n) = the number of relaxed compacted binary trees of right height at most one of size n. A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The number of unbounded relaxed compacted binary trees of size n is A082161(n). See the Genitrini et al. link. - Michael Wallner, Jun 20 2017
Also the number of distinct adjacency matrices in the n-ladder rung graph. - Eric W. Weisstein, Jul 22 2017
From Christopher J. Smyth, Jan 26 2018: (Start)
a(n) = the number of essentially different ways of writing a probability distribution taking n+1 values as a sum of products of binary probability distributions. See comment of Mitch Harris above. This is because each such way corresponds to a full binary tree with n+1 leaves, with the leaves labeled by the values. (This comment is due to Niko Brummer.)
Also the number of binary trees with root labeled by an (n+1)-set S, its n+1 leaves by the singleton subsets of S, and other nodes labeled by subsets T of S so that the two daughter nodes of the node labeled by T are labeled by the two parts of a 2-partition of T. This also follows from Mitch Harris' comment above, since the leaf labels determine the labels of the other vertices of the tree.
(End)
a(n) is the n-th moment of the chi-squared distribution with one degree of freedom (equivalent to Coleman's Apr 06 2004 comment). - Bryan R. Gillespie, Mar 07 2021
Let b(n) = 0 for n odd and b(2k) = a(k); i.e., let the sequence b(n) be an aerated version of this entry. After expanding the differential operator (x + D)^n and normal ordering the resulting terms, the integer coefficient of the term x^k D^m is n! b(n-k-m) / [(n-k-m)! k! m!] with 0 <= k,m <= n and (k+m) <= n. E.g., (x+D)^2 = x^2 + 2xD + D^2 + 1 with D = d/dx. The result generalizes to the raising (R) and lowering (L) operators of any Sheffer polynomial sequence by replacing x by R and D by L and follows from the disentangling relation e^{t(L+R)} = e^{t^2/2} e^{tR} e^{tL}. Consequently, these are also the coefficients of the reordered 2^n permutations of the binary symbols L and R under the condition LR = RL + 1. E.g., (L+R)^2 = LL + LR + RL + RR = LL + 2RL + RR + 1. (Cf. A344678.) - Tom Copeland, May 25 2021
From Tom Copeland, Jun 14 2021: (Start)
Lando and Zvonkin present several scenarios in which the double factorials occur in their role of enumerating perfect matchings (pairings) and as the nonzero moments of the Gaussian e^(x^2/2).
Speyer and Sturmfels (p. 6) state that the number of facets of the abstract simplicial complex known as the tropical Grassmannian G'''(2,n), the space of phylogenetic T_n trees (see A134991), or Whitehouse complex is a shifted double factorial.
These are also the unsigned coefficients of the x[2]^m terms in the partition polynomials of A134685 for compositional inversion of e.g.f.s, a refinement of A134991.
a(n)*2^n = A001813(n) and A001813(n)/(n+1)! = A000108(n), the Catalan numbers, the unsigned coefficients of the x[2]^m terms in the partition polynomials A133437 for compositional inversion of o.g.f.s, a refinement of A033282, A126216, and A086810. Then the double factorials inherit a multitude of analytic and combinatoric interpretations from those of the Catalan numbers, associahedra, and the noncrossing partitions of A134264 with the Catalan numbers as unsigned-row sums. (End)
Connections among the Catalan numbers A000108, the odd double factorials, values of the Riemann zeta function and its derivative for integer arguments, and series expansions of the reduced action for the simple harmonic oscillator and the arc length of the spiral of Archimedes are given in the MathOverflow post on the Riemann zeta function. - Tom Copeland, Oct 02 2021
b(n) = a(n) / (n! 2^n) = Sum_{k = 0..n} (-1)^n binomial(n,k) (-1)^k a(k) / (k! 2^k) = (1-b.)^n, umbrally; i.e., the normalized double factorial a(n) is self-inverse under the binomial transform. This can be proved by applying the Euler binomial transformation for o.g.f.s Sum_{n >= 0} (1-b.)^n x^n = (1/(1-x)) Sum_{n >= 0} b_n (x / (x-1))^n to the o.g.f. (1-x)^{-1/2} = Sum_{n >= 0} b_n x^n. Other proofs are suggested by the discussion in Watson on pages 104-5 of transformations of the Bessel functions of the first kind with b(n) = (-1)^n binomial(-1/2,n) = binomial(n-1/2,n) = (2n)! / (n! 2^n)^2. - Tom Copeland, Dec 10 2022

Examples

			a(3) = 1*3*5 = 15.
From _Joerg Arndt_, Sep 10 2013: (Start)
There are a(3)=15 involutions of 6 elements without fixed points:
  #:    permutation           transpositions
  01:  [ 1 0 3 2 5 4 ]      (0, 1) (2, 3) (4, 5)
  02:  [ 1 0 4 5 2 3 ]      (0, 1) (2, 4) (3, 5)
  03:  [ 1 0 5 4 3 2 ]      (0, 1) (2, 5) (3, 4)
  04:  [ 2 3 0 1 5 4 ]      (0, 2) (1, 3) (4, 5)
  05:  [ 2 4 0 5 1 3 ]      (0, 2) (1, 4) (3, 5)
  06:  [ 2 5 0 4 3 1 ]      (0, 2) (1, 5) (3, 4)
  07:  [ 3 2 1 0 5 4 ]      (0, 3) (1, 2) (4, 5)
  08:  [ 3 4 5 0 1 2 ]      (0, 3) (1, 4) (2, 5)
  09:  [ 3 5 4 0 2 1 ]      (0, 3) (1, 5) (2, 4)
  10:  [ 4 2 1 5 0 3 ]      (0, 4) (1, 2) (3, 5)
  11:  [ 4 3 5 1 0 2 ]      (0, 4) (1, 3) (2, 5)
  12:  [ 4 5 3 2 0 1 ]      (0, 4) (1, 5) (2, 3)
  13:  [ 5 2 1 4 3 0 ]      (0, 5) (1, 2) (3, 4)
  14:  [ 5 3 4 1 2 0 ]      (0, 5) (1, 3) (2, 4)
  15:  [ 5 4 3 2 1 0 ]      (0, 5) (1, 4) (2, 3)
(End)
G.f. = 1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + 135135*x^7 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, (26.2.28).
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 317.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 228, #19.
  • Hoel, Port and Stone, Introduction to Probability Theory, Section 7.3.
  • F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem, North-Holland, 1992, see p. 14.
  • C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.6 and also p. 178.
  • R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer-Verlag, New York, 1999, p. 73.
  • G. Watson, The Theory of Bessel Functions, Cambridge Univ. Press, 1922.

Links

Crossrefs

Cf. A000085, A006882, A000165 ((2n)!!), A001818, A009445, A039683, A102992, A001190 (no labels), A000680, A132101.
Cf. A086677; A055142 (for this sequence, |a(n+1)| + 1 is the number of distinct products which can be formed using commutative, nonassociative multiplication and a nonempty subset of n given variables).
Constant terms of polynomials in A098503. First row of array A099020.
Cf. A079267, A000698, A029635, A161198, A076795, A123023, A161124, A051125, A181983, A099174, A087547, A028338 (first column).
Subsequence of A248652.
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A053871 (binomial transform).
Cf. A000108, A001813, A033282, A060540, A086810, A094638, A126216, A133437, A134264, A134685, A134991, A344678.

Programs

  • GAP
    A001147 := function(n) local i, s, t; t := 1; i := 0; Print(t, ", "); for i in [1 .. n] do t := t*(2*i-1); Print(t, ", "); od; end; A001147(100); # Stefano Spezia, Nov 13 2018
  • Haskell
    a001147 n = product [1, 3 .. 2 * n - 1]
a001147_list = 1 : zipWith (*) [1, 3 ..] a001147_list
-- Reinhard Zumkeller, Feb 15 2015, Dec 03 2011
  • Magma
    A001147:=func< n | n eq 0 select 1 else &*[ k: k in [1..2*n-1 by 2] ] >; [ A001147(n): n in [0..20] ]; // Klaus Brockhaus, Jun 22 2011
  • Magma
    I:=[1,3]; [1] cat [n le 2 select I[n]  else (3*n-2)*Self(n-1)-(n-1)*(2*n-3)*Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Feb 19 2015
  • Maple
    f := n->(2*n)!/(n!*2^n);
A001147 := proc(n) doublefactorial(2*n-1); end: # R. J. Mathar, Jul 04 2009
A001147 := n -> 2^n*pochhammer(1/2, n); # Peter Luschny, Aug 09 2009
G(x):=(1-2*x)^(-1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # Zerinvary Lajos, Apr 03 2009; aligned with offset by Johannes W. Meijer, Aug 11 2009
series(hypergeom([1,1/2],[],2*x),x=0,20); # Mark van Hoeij, Apr 07 2013
  • Mathematica
    Table[(2 n - 1)!!, {n, 0, 19}] (* Robert G. Wilson v, Oct 12 2005 *)
a[ n_] := 2^n Gamma[n + 1/2] / Gamma[1/2]; (* Michael Somos, Sep 18 2014 *)
Join[{1}, Range[1, 41, 2]!!] (* Harvey P. Dale, Jan 28 2017 *)
a[ n_] := If[ n < 0, (-1)^n / a[-n], SeriesCoefficient[ Product[1 - (1 - x)^(2 k - 1), {k, n}], {x, 0, n}]]; (* Michael Somos, Jun 27 2017 *)
(2 Range[0, 20] - 1)!! (* Eric W. Weisstein, Jul 22 2017 *)
  • Maxima
    a(n):=if n=0 then 1 else sum(sum(binomial(n-1,i)*binomial(n-i-1,j)*a(i)*a(j)*a(n-i-j-1),j,0,n-i-1),i,0,n-1); /* Vladimir Kruchinin, May 06 2020 */
  • PARI
    {a(n) = if( n<0, (-1)^n / a(-n), (2*n)! / n! / 2^n)}; /* Michael Somos, Sep 18 2014 */
  • PARI
    x='x+O('x^33); Vec(serlaplace((1-2*x)^(-1/2))) \\ Joerg Arndt, Apr 24 2011
  • Python
    from sympy import factorial2
def a(n): return factorial2(2 * n - 1)
print([a(n) for n in range(101)])  # Indranil Ghosh, Jul 22 2017
  • Sage
    [rising_factorial(n+1,n)/2^n for n in (0..15)] # Peter Luschny, Jun 26 2012

Formula

E.g.f.: 1 / sqrt(1 - 2*x).
D-finite with recurrence: a(n) = a(n-1)*(2*n-1) = (2*n)!/(n!*2^n) = A010050(n)/A000165(n).
a(n) ~ sqrt(2) * 2^n * (n/e)^n.
Rational part of numerator of Gamma(n+1/2): a(n) * sqrt(Pi) / 2^n = Gamma(n+1/2). - Yuriy Brun, Ewa Dominowska (brun(AT)mit.edu), May 12 2001
With interpolated zeros, the sequence has e.g.f. exp(x^2/2). - Paul Barry, Jun 27 2003
The Ramanujan polynomial psi(n+1, n) has value a(n). - Ralf Stephan, Apr 16 2004
a(n) = Sum_{k=0..n} (-2)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
Log(1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + ...) = x + 5/2*x^2 + 37/3*x^3 + 353/4*x^4 + 4081/5*x^5 + 55205/6*x^6 + ..., where [1, 5, 37, 353, 4081, 55205, ...] = A004208. - Philippe Deléham, Jun 20 2006
1/3 + 2/15 + 3/105 + ... = 1/2. [Jolley eq. 216]
Sum_{j=1..n} j/a(j+1) = (1 - 1/a(n+1))/2. [Jolley eq. 216]
1/1 + 1/3 + 2/15 + 6/105 + 24/945 + ... = Pi/2. - Gary W. Adamson, Dec 21 2006
a(n) = (1/sqrt(2*Pi))*Integral_{x>=0} x^n*exp(-x/2)/sqrt(x). - Paul Barry, Jan 28 2008
a(n) = A006882(2n-1). - R. J. Mathar, Jul 04 2009
G.f.: 1/(1-x-2x^2/(1-5x-12x^2/(1-9x-30x^2/(1-13x-56x^2/(1- ... (continued fraction). - Paul Barry, Sep 18 2009
a(n) = (-1)^n*subs({log(e)=1,x=0},coeff(simplify(series(e^(x*t-t^2/2),t,2*n+1)),t^(2*n))*(2*n)!). - Leonid Bedratyuk, Oct 31 2009
a(n) = 2^n*gamma(n+1/2)/gamma(1/2). - Jaume Oliver Lafont, Nov 09 2009
G.f.: 1/(1-x/(1-2x/(1-3x/(1-4x/(1-5x/(1- ...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009
The g.f. of a(n+1) is 1/(1-3x/(1-2x/(1-5x/(1-4x/(1-7x/(1-6x/(1-.... (continued fraction). - Paul Barry, Dec 04 2009
a(n) = Sum_{i=1..n} binomial(n,i)*a(i-1)*a(n-i). - Vladimir Shevelev, Sep 30 2010
E.g.f.: A(x) = 1 - sqrt(1-2*x) satisfies the differential equation A'(x) - A'(x)*A(x) - 1 = 0. - Vladimir Kruchinin, Jan 17 2011
a(n) = A123023(2*n). - Michael Somos, Jul 24 2011
a(n) = (1/2)*Sum_{i=1..n} binomial(n+1,i)*a(i-1)*a(n-i). See link above. - Dennis P. Walsh, Dec 02 2011
a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n,n+k)*Stirling_1(n+k,k) [Kauers and Ko].
a(n) = A035342(n, 1), n >= 1 (first column of triangle).
a(n) = A001497(n, 0) = A001498(n, n), first column, resp. main diagonal, of Bessel triangle.
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n and sum of top row terms of M^(n-1), where M = a variant of the (1,2) Pascal triangle (Cf. A029635) as the following production matrix:
1, 2, 0, 0, 0, ...
1, 3, 2, 0, 0, ...
1, 4, 5, 2, 0, ...
1, 5, 9, 7, 2, ...
...
For example, a(3) = 15 is the left term in top row of M^3: (15, 46, 36, 8) and a(4) = 105 = (15 + 46 + 36 + 8).
(End)
G.f.: A(x) = 1 + x/(W(0) - x); W(k) = 1 + x + x*2*k - x*(2*k + 3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
a(n) = Sum_{i=1..n} binomial(n,i-1)*a(i-1)*a(n-i). - Dennis P. Walsh, Dec 02 2011
a(n) = A009445(n) / A014481(n). - Reinhard Zumkeller, Dec 03 2011
a(n) = (-1)^n*Sum_{k=0..n} 2^(n-k)*s(n+1,k+1), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = (2*n)4! = Gauss_factorial(2*n,4) = Product{j=1..2*n, gcd(j,4)=1} j. - Peter Luschny, Oct 01 2012
G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(2*k - 1)/(1 - x*(2*k + 2)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: 1 + x/Q(0), where Q(k) = 1 + (2*k - 1)*x - 2*x*(k + 1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) - 1 + 2*x*(2*k + 2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(2*k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: G(0), where G(k) = 1 + 2*x*(4*k + 1)/(4*k + 2 - 2*x*(2*k + 1)*(4*k + 3)/(x*(4*k + 3) + 2*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013
a(n) = (2*n - 3)*a(n-2) + (2*n - 2)*a(n-1), n > 1. - Ivan N. Ianakiev, Jul 08 2013
G.f.: G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 04 2013
a(n) = 2*a(n-1) + (2n-3)^2*a(n-2), a(0) = a(1) = 1. - Philippe Deléham, Oct 27 2013
G.f. of reciprocals: Sum_{n>=0} x^n/a(n) = 1F1(1; 1/2; x/2), confluent hypergeometric Function. - R. J. Mathar, Jul 25 2014
0 = a(n)*(+2*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) for all n in Z. - Michael Somos, Sep 18 2014
a(n) = (-1)^n / a(-n) = 2*a(n-1) + a(n-1)^2 / a(n-2) for all n in Z. - Michael Somos, Sep 18 2014
From Peter Bala, Feb 18 2015: (Start)
Recurrence equation: a(n) = (3*n - 2)*a(n-1) - (n - 1)*(2*n - 3)*a(n-2) with a(1) = 1 and a(2) = 3.
The sequence b(n) = A087547(n), beginning [1, 4, 52, 608, 12624, ... ], satisfies the same second-order recurrence equation. This leads to the generalized continued fraction expansion lim_{n -> infinity} b(n)/a(n) = Pi/2 = 1 + 1/(3 - 6/(7 - 15/(10 - ... - n*(2*n - 1)/((3*n + 1) - ... )))). (End)
E.g.f of the sequence whose n-th element (n = 1,2,...) equals a(n-1) is 1-sqrt(1-2*x). - Stanislav Sykora, Jan 06 2017
Sum_{n >= 1} a(n)/(2*n-1)! = exp(1/2). - Daniel Suteu, Feb 06 2017
a(n) = A028338(n, 0), n >= 0. - Wolfdieter Lang, May 27 2017
a(n) = (Product_{k=0..n-2} binomial(2*(n-k),2))/n!. - Stefano Spezia, Nov 13 2018
a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} C(n-1,i)*C(n-i-1,j)*a(i)*a(j)*a(n-i-j-1), a(0)=1, - Vladimir Kruchinin, May 06 2020
From Amiram Eldar, Jun 29 2020: (Start)
Sum_{n>=1} 1/a(n) = sqrt(e*Pi/2)*erf(1/sqrt(2)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erfi is the imaginary error function. (End)
G.f. of reciprocals: R(x) = Sum_{n>=0} x^n/a(n) satisfies (1 + x)*R(x) = 1 + 2*x*R'(x). - Werner Schulte, Nov 04 2024

Extensions

Removed erroneous comments: neither the number of n X n binary matrices A such that A^2 = 0 nor the number of simple directed graphs on n vertices with no directed path of length two are counted by this sequence (for n = 3, both are 13). - Dan Drake, Jun 02 2009

A000096 a(n) = n*(n+3)/2.

Original entry on oeis.org

0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377, 1430, 1484, 1539, 1595, 1652, 1710, 1769
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For n >= 1, a(n) is the maximal number of pieces that can be obtained by cutting an annulus with n cuts. See illustration. - Robert G. Wilson v
n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
n(n-3)/2 (n >= 4) is the degree of the third-smallest irreducible presentation of the symmetric group S_n (cf. James and Kerber, Appendix 1).
a(n) is also the multiplicity of the eigenvalue (-2) of the triangle graph Delta(n+1). (See p. 19 in Biggs.) - Felix Goldberg (felixg(AT)tx.technion.ac.il), Nov 25 2001
For n > 3, a(n-3) = dimension of the traveling salesman polytope T(n). - Benoit Cloitre, Aug 18 2002
Also counts quasi-dominoes (quasi-2-ominoes) on an n X n board. Cf. A094170-A094172. - Jon Wild, May 07 2004
Coefficient of x^2 in (1 + x + 2*x^2)^n. - Michael Somos, May 26 2004
a(n) is the number of "prime" n-dimensional polyominoes. A "prime" n-polyomino cannot be formed by connecting any other n-polyominoes except for the n-monomino and the n-monomino is not prime. E.g., for n=1, the 1-monomino is the line of length 1 and the only "prime" 1-polyominoes are the lines of length 2 and 3. This refers to "free" n-dimensional polyominoes, i.e., that can be rotated along any axis. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005
Solutions to the quadratic equation q(m, r) = (-3 +- sqrt(9 + 8(m - r))) / 2, where m - r is included in a(n). Let t(m) = the triangular number (A000217) less than some number k and r = k - t(m). If k is neither prime nor a power of two and m - r is included in A000096, then m - q(m, r) will produce a value that shares a divisor with k. - Andrew S. Plewe, Jun 18 2005
Sum_{k=2..n+1} 4/(k*(k+1)*(k-1)) = ((n+3)*n)/((n+2)*(n+1)). Numerator(Sum_{k=2..n+1} 4/(k*(k+1)*(k-1))) = (n+3)*n/2. - Alexander Adamchuk, Apr 11 2006
Number of rooted trees with n+3 nodes of valence 1, no nodes of valence 2 and exactly two other nodes. I.e., number of planted trees with n+2 leaves and exactly two branch points. - Theo Johnson-Freyd (theojf(AT)berkeley.edu), Jun 10 2007
If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-2)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
For n >= 1, a(n) is the number of distinct shuffles of the identity permutation on n+1 letters with the identity permutation on 2 letters (12). - Camillia Smith Barnes, Oct 04 2008
If s(n) is a sequence defined as s(1) = x, s(n) = kn + s(n-1) + p for n > 1, then s(n) = a(n-1)*k + (n-1)*p + x. - Gary Detlefs, Mar 04 2010
The only primes are a(1) = 2 and a(2) = 5. - Reinhard Zumkeller, Jul 18 2011
a(n) = m such that the (m+1)-th triangular number minus the m-th triangular number is the (n+1)-th triangular number: (m+1)(m+2)/2 - m(m+1)/2 = (n+1)(n+2)/2. - Zak Seidov, Jan 22 2012
For n >= 1, number of different values that Sum_{k=1..n} c(k)*k can take where the c(k) are 0 or 1. - Joerg Arndt, Jun 24 2012
On an n X n chessboard (n >= 2), the number of possible checkmate positions in the case of king and rook versus a lone king is 0, 16, 40, 72, 112, 160, 216, 280, 352, ..., which is 8*a(n-2). For a 4 X 4 board the number is 40. The number of positions possible was counted including all mirror images and rotations for all four sides of the board. - Jose Abutal, Nov 19 2013
If k = a(i-1) or k = a(i+1) and n = k + a(i), then C(n, k-1), C(n, k), C(n, k+1) are three consecutive binomial coefficients in arithmetic progression and these are all the solutions. There are no four consecutive binomial coefficients in arithmetic progression. - Michael Somos, Nov 11 2015
a(n-1) is also the number of independent components of a symmetric traceless tensor of rank 2 and dimension n >= 1. - Wolfdieter Lang, Dec 10 2015
Numbers k such that 8k + 9 is a square. - Juri-Stepan Gerasimov, Apr 05 2016
Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See the Wojnar et al. link] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N)= ((-1)^D/(D-1)!)*(D-N)*N^chi*g_D(N) where chi = (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). Then a(n) are the negated coefficients of the next to the highest order term in the polynomials N^chi*g_D(N), starting at D=3. - Gregory Gerard Wojnar, Jul 19 2017
For n >= 2, a(n) is the number of summations required to solve the linear regression of n variables (n-1 independent variables and 1 dependent variable). - Felipe Pedraza-Oropeza, Dec 07 2017
For n >= 2, a(n) is the number of sums required to solve the linear regression of n variables: 5 for two variables (sums of X, Y, X^2, Y^2, X*Y), 9 for 3 variables (sums of X1, X2, Y1, X1^2, X1*X2, X1*Y, X2^2, X2*Y, Y^2), and so on. - Felipe Pedraza-Oropeza, Jan 11 2018
a(n) is the area of a triangle with vertices at (n, n+1), ((n+1)*(n+2)/2, (n+2)*(n+3)/2), ((n+2)^2, (n+3)^2). - J. M. Bergot, Jan 25 2018
Number of terms less than 10^k: 1, 4, 13, 44, 140, 446, 1413, 4471, 14141, 44720, 141420, 447213, ... - Muniru A Asiru, Jan 25 2018
a(n) is also the number of irredundant sets in the (n+1)-path complement graph for n > 2. - Eric W. Weisstein, Apr 11 2018
a(n) is also the largest number k such that the largest Dyck path of the symmetric representation of sigma(k) has exactly n peaks, n >= 1. (Cf. A237593.) - Omar E. Pol, Sep 04 2018
For n > 0, a(n) is the number of facets of associahedra. Cf. A033282 and A126216 and their refinements A111785 and A133437 for related combinatorial and analytic constructs. See p. 40 of Hanson and Sha for a relation to projective spaces and string theory. - Tom Copeland, Jan 03 2021
For n > 0, a(n) is the number of bipartite graphs with 2n or 2n+1 edges, no isolated vertices, and a stable set of cardinality 2. - Christian Barrientos, Jun 13 2022
For n >= 2, a(n-2) is the number of permutations in S_n which are the product of two different transpositions of adjacent points. - Zbigniew Wojciechowski, Mar 31 2023
a(n) represents the optimal stop-number to achieve the highest running score for the Greedy Pig game with an (n-1)-sided die with a loss on a 1. The total at which one should stop is a(s-1), e.g. for a 6-sided die, one should pass the die at 20. See Sparks and Haran. - Nicholas Stefan Georgescu, Jun 09 2024

Examples

			G.f. = 2*x + 5*x^2 + 9*x^3 + 14*x^4 + 20*x^5 + 27*x^6 + 35*x^7 + 44*x^8 + 54*x^9 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
  • Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.
  • G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Maths. and its Appls., Vol. 16, Addison-Wesley, 1981, Reading, MA, U.S.A.
  • D. G. Kendall et al., Shape and Shape Theory, Wiley, 1999; see p. 4.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Complement of A007401. Column 2 of A145324. Column of triangle A014473, first skew subdiagonal of A033282, a diagonal of A079508.
Occurs as a diagonal in A074079/A074080, i.e., A074079(n+3, n) = A000096(n-1) for all n >= 2. Also A074092(n) = 2^n * A000096(n-1) after n >= 2.
Cf. A000124, A000217, A005581-A005584, A023531, A034856, A067550, A127672 (Chebyshev C).
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488.
Similar sequences are listed in A316466.
Cf. A033282, A111785, A126216, A133437.

Programs

Formula

G.f.: A(x) = x*(2-x)/(1-x)^3. a(n) = binomial(n+1, n-1) + binomial(n, n-1).
Connection with triangular numbers: a(n) = A000217(n+1) - 1.
a(n) = a(n-1) + n + 1. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005
a(n) = 2*t(n) - t(n-1) where t() are the triangular numbers, e.g., a(5) = 2*t(5) - t(4) = 2*15 - 10 = 20. - Jon Perry, Jul 23 2003
a(-3-n) = a(n). - Michael Somos, May 26 2004
2*a(n) = A008778(n) - A105163(n). - Creighton Dement, Apr 15 2005
a(n) = C(3+n, 2) - C(3+n, 1). - Zerinvary Lajos, Dec 09 2005
a(n) = A067550(n+1) / A067550(n). - Alexander Adamchuk, May 20 2006
a(n) = A126890(n,1) for n > 0. - Reinhard Zumkeller, Dec 30 2006
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Paul Curtz, Jan 02 2008
Starting (2, 5, 9, 14, ...) = binomial transform of (2, 3, 1, 0, 0, 0, ...). - Gary W. Adamson, Jul 03 2008
For n >= 0, a(n+2) = b(n+1) - b(n), where b(n) is the sequence A005586. - K.V.Iyer, Apr 27 2009
A002262(a(n)) = n. - Reinhard Zumkeller, May 20 2009
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n-1)=coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jul 08 2010
a(n) = Sum_{k=1..n} (k+1)!/k!. - Gary Detlefs, Aug 03 2010
a(n) = n(n+1)/2 + n = A000217(n) + n. - Zak Seidov, Jan 22 2012
E.g.f.: F(x) = 1/2*x*exp(x)*(x+4) satisfies the differential equation F''(x) - 2*F'(x) + F(x) = exp(x). - Peter Bala, Mar 14 2012
a(n) = binomial(n+3, 2) - (n+3). - Robert G. Wilson v, Mar 15 2012
a(n) = A181971(n+1, 2) for n > 0. - Reinhard Zumkeller, Jul 09 2012
a(n) = A214292(n+2, 1). - Reinhard Zumkeller, Jul 12 2012
G.f.: -U(0) where U(k) = 1 - 1/((1-x)^2 - x*(1-x)^4/(x*(1-x)^2 - 1/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 27 2012
A023532(a(n)) = 0. - Reinhard Zumkeller, Dec 04 2012
a(n) = A014132(n,n) for n > 0. - Reinhard Zumkeller, Dec 12 2012
a(n-1) = (1/n!)*Sum_{j=0..n} binomial(n,j)*(-1)^(n-j)*j^n*(j-1). - Vladimir Kruchinin, Jun 06 2013
a(n) = 2n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=2..n+1} i. - Wesley Ivan Hurt, Jun 28 2013
Sum_{n>0} 1/a(n) = 11/9. - Enrique Pérez Herrero, Nov 26 2013
a(n) = Sum_{i=1..n} (n - i + 2). - Wesley Ivan Hurt, Mar 31 2014
A023531(a(n)) = 1. - Reinhard Zumkeller, Feb 14 2015
For n > 0: a(n) = A101881(2*n-1). - Reinhard Zumkeller, Feb 20 2015
a(n) + a(n-1) = A008865(n+1) for all n in Z. - Michael Somos, Nov 11 2015
a(n+1) = A127672(4+n, n), n >= 0, where A127672 gives the coefficients of the Chebyshev C polynomials. See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015
a(n) = (n+1)^2 - A000124(n). - Anton Zakharov, Jun 29 2016
Dirichlet g.f.: (zeta(s-2) + 3*zeta(s-1))/2. - Ilya Gutkovskiy, Jun 30 2016
a(n) = 2*A000290(n+3) - 3*A000217(n+3). - J. M. Bergot, Apr 04 2018
a(n) = Stirling2(n+2, n+1) - 1. - Peter Luschny, Jan 05 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/3 - 5/9. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = 3.
Product_{n>=1} (1 - 1/a(n)) = 3*cos(sqrt(17)*Pi/2)/(4*Pi). (End)
Product_{n>=0} a(4*n+1)*a(4*n+4)/(a(4*n+2)*a(4*n+3)) = Pi/6. - Michael Jodl, Apr 05 2025

A001003 Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.

Original entry on oeis.org

1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869, 71039373, 372693519, 1968801519, 10463578353, 55909013009, 300159426963, 1618362158587, 8759309660445, 47574827600981, 259215937709463, 1416461675464871
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

If you are looking for the Schroeder numbers (a.k.a. large Schroder numbers, or big Schroeder numbers), see A006318.
Yang & Jiang (2021) call these the small 2-Schroeder numbers. - N. J. A. Sloane, Mar 28 2021
There are two schools of thought about the index for the first term. I prefer the indexing a(0) = a(1) = 1, a(2) = 3, a(3) = 11, etc.
a(n) is the number of ways to insert parentheses in a string of n+1 symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored.
Also length of list produced by a variant of the Catalan producing iteration: replace each integer k with the list 0,1,..,k,k+1,k,...,1,0 and get the length a(n) of the resulting (flattened) list after n iterations. - Wouter Meeussen, Nov 11 2001
Stanley gives several other interpretations for these numbers.
Number of Schroeder paths of semilength n (i.e., lattice paths from (0,0) to (2n,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no peaks at level 1. Example: a(2)=3 because among the six Schroeder paths of semilength two HH, UHD, UUDD, HUD, UDH and UDUD, only the first three have no peaks at level 1. - Emeric Deutsch, Dec 27 2003
a(n+1) is the number of Dyck n-paths in which the interior vertices of the ascents are colored white or black. - David Callan, Mar 14 2004
Number of possible schedules for n time slots in the first-come first-served (FCFS) printer policy.
Also row sums of A086810, A033282, A126216. - Philippe Deléham, May 09 2004
a(n+1) is the number of pairs (u,v) of same-length compositions of n, 0's allowed in u but not in v and u dominates v (meaning u_1 >= v_1, u_1 + u_2 >= v_1 + v_2 and so on). For example, with n=2, a(3) counts (2,2), (1+1,1+1), (2+0,1+1). - David Callan, Jul 20 2005
The big Schroeder number (A006318) is the number of Schroeder paths from (0,0) to (n,n) (subdiagonal paths with steps (1,0) (0,1) and (1,1)). These paths fall in two classes: those with steps on the main diagonal and those without. These two classes are equinumerous and the number of paths in either class is the little Schroeder number a(n) (half the big Schroeder number). - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005
With offset 0, a(n) = number of (colored) Motzkin (n-1)-paths with each upstep U getting one of 2 colors and each flatstep F getting one of 3 colors. Example. With their colors immediately following upsteps/flatsteps, a(2) = 3 counts F1, F2, F3 and a(3)=11 counts U1D, U2D, F1F1, F1F2, F1F3, F2F1, F2F2, F2F3, F3F1, F3F2, F3F3. - David Callan, Aug 16 2006
Shifts left when INVERT transform applied twice. - Alois P. Heinz, Apr 01 2009
Number of increasing tableaux of shape (n,n). An increasing tableau is a semistandard tableaux with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 3 because of the three tableaux (12)(34), (13)(24), (12)(23). - Oliver Pechenik, Apr 22 2014
Number of ordered trees with no vertex of outdegree 1 and having n+1 leaves (called sometimes Schröder trees). - Emeric Deutsch, Dec 13 2014
Number of dissections of a convex (n+2)-gon by nonintersecting diagonals. Example: a(2)=3 because for a square ABCD we have (i) no diagonal, (ii) dissection with diagonal AC, and (iii) dissection with diagonal BD. - Emeric Deutsch, Dec 13 2014
The little Schroeder numbers are the moments of the Marchenko-Pastur law for the case c=2 (although the moment m0 is 1/2 instead of 1): 1/2, 1, 3, 11, 45, 197, 903, ... - Jose-Javier Martinez, Apr 07 2015
Number of generalized Motzkin paths with no level steps at height 0, from (0,0) to (2n,0), and consisting of steps U=(1,1), D=(1,-1) and H2=(2,0). For example, for n=3, we have the 11 paths: UDUDUD, UUDDUD, UDUUDD, UH2DUD, UDUH2D, UH2H2D, UUDUDD, UUUDDD, UUH2DD, UUDH2D, UH2UDD. - José Luis Ramírez Ramírez, Apr 20 2015
REVERT transform of A225883. - Vladimir Reshetnikov, Oct 25 2015
Total number of (nonempty) faces of all dimensions in the associahedron K_{n+1} of dimension n-1. For example, K_4 (a pentagon) includes 5 vertices and 5 edges together with K_4 itself (5 + 5 + 1 = 11), while K_5 includes 14 vertices, 21 edges and 9 faces together with K_5 itself (14 + 21 + 9 + 1 = 45). - Noam Zeilberger, Sep 17 2018
a(n) is the number of interval posets of permutations with n minimal elements that have exactly two realizers, up to a shift by 1 in a(4). See M. Bouvel, L. Cioni, B. Izart, Theorem 17 page 13. - Mathilde Bouvel, Oct 21 2021
a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_1 = 1, (ii) u_i <= i for all i, (iii) the nonzero u_i are weakly increasing. For example, a(2) = 3 counts 10, 11, 12, and a(3) = 11 counts 100, 101, 102, 103, 110, 111, 112, 113, 120, 122, 123. See link below. - David Callan, Dec 19 2021
a(n) is the number of parking functions of size n avoiding the patterns 132 and 213. - Lara Pudwell, Apr 10 2023
a(n+1) is the number of Schroeder paths from (0,0) to (2n,0) in which level steps at height 0 come in 2 colors. - Alexander Burstein, Jul 23 2023

Examples

			G.f. = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + 197*x^5 + 903*x^6 + 4279*x^7 + ...
a(2) = 3: abc, a(bc), (ab)c; a(3) = 11: abcd, (ab)cd, a(bc)d, ab(cd), (ab)(cd), a(bcd), a(b(cd)), a((bc)d), (abc)d, (a(bc))d, ((ab)c)d.
Sum over partitions formula: a(3) = 2*a(0)*a(2) + 1*a(1)^2 + 3*(a(0)^2)*a(1) + 1*a(0)^4 = 6 + 1 + 3 + 1 = 11.
a(4) = 45 since the top row of Q^3 = (11, 14, 12, 8, 0, 0, 0, ...); (11 + 14 + 12 + 8) = 45.

References

  • D. Arques and A. Giorgetti, Une bijection géometrique entre une famille d'hypercartes et une famille de polygones énumérées par la série de Schroeder, Discrete Math., 217 (2000), 17-32.
  • Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
  • N. Bernasconi et al., On properties of random dissections and triangulations, Combinatorica, 30 (6) (2010), 627-654.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 618.
  • Peter J. Cameron, Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155, also p. 179, line -9. - N. J. A. Sloane, Apr 18 2014
  • C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 57.
  • D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From N. J. A. Sloane, May 11 2012
  • Emeric Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.
  • Tomislav Doslic and Darko Veljan, Logarithmic behavior of some combinatorial sequences. Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - From N. J. A. Sloane, May 01 2012
  • Michael Drmota, Anna de Mier, and Marc Noy, Extremal statistics on non-crossing configurations. Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (2). - N. J. A. Sloane, May 18 2014
  • I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
  • I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. Part II is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.
  • P. Flajolet and M. Noy, Analytic combinatorics of non-crossing permutations, Discrete Math., 204 (1999), 203-229, Section 3.1.
  • D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence, J. Comb Thy A 80 380-384 1997.
  • Wolfgang Gatterbauer and Dan Suciu, Dissociation and propagation for approximate lifted inference with standard relational database management systems, The VLDB Journal, February 2017, Volume 26, Issue 1, pp. 5-30; DOI 10.1007/s00778-016-0434-5
  • Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
  • D. Gouyou-Beauchamps and B. Vauquelin, Deux propriétés combinatoires des nombres de Schroeder, Theor. Inform. Appl., 22 (1988), 361-388.
  • N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
  • P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.
  • M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.
  • D. E. Knuth, The Art of Computer Programming, Vol. 1, various sections (e.g. p. 534 of 2nd ed.).
  • D. E. Knuth, The Art of Computer Programming, Vol. 1, (p. 539 of 3rd ed.).
  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.2.1.6, Problem 66, p. 479.
  • J. S. Lew, Polynomial enumeration of multidimensional lattices, Math. Systems Theory, 12 (1979), 253-270.
  • Ana Marco and J.-J. Martinez, A total positivity property of the Marchenko-Pastur Law, Electronic Journal of Linear Algebra, 30 (2015), #7.
  • T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
  • L. Ozsvart, Counting ordered graphs that avoid certain subgraphs, Discr. Math., 339 (2016), 1871-1877.
  • R. C. Read, On general dissections of a polygon, Aequat. Mathem. 18 (1978) 370-388, Table 6
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 168.
  • E. Schroeder, Vier combinatorische Probleme, Zeit. f. Math. Phys., 15 (1870), 361-376.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178; see page 239, Exercise 6.39b.
  • H. N. V. Temperley and D. G. Rogers, A note on Baxter's generalization of the Temperley-Lieb operators, pp. 324-328 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.
  • I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 198.
  • Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

Links

Crossrefs

See A000081, A000108, A001190, A001699, for other ways to count parentheses.
Row sums of A033282, A033877, A086810, A126216.
Right-hand column 1 of convolution triangle A011117.
Column 1 of A336573. Column 0 of A104219.
The sequences listed in Yang-Jiang's Table 1 appear to be A006318, this sequence, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021
Cf. A000041, A000311, A001263, A003480, A006125, A010683, A025240, A035011.
Cf. A048996, A054726, A059435, A065096, A080243, A085403, A086456, A086616.
Cf. A086810, A088617, A104858, A110440, A111785, A114710, A118376, A126216.
Cf. A139685, A144156, A144944, A153881, A186826.
Cf. A006318 (Schroeder numbers).

Programs

  • Haskell
    a001003 = last . a144944_row  -- Reinhard Zumkeller, May 11 2013
  • Magma
    R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!( (1+x -Sqrt(1-6*x+x^2) )/(4*x) )); // G. C. Greubel, Oct 27 2024
  • Maple
    t1 := (1/(4*x))*(1+x-sqrt(1-6*x+x^2)); series(t1,x,40);
invtr:= proc(p) local b; b:= proc(n) option remember; local i; `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1)) end end: a:='a': f:= (invtr@@2)(a): a:= proc(n) if n<0 then 1 else f(n-1) fi end: seq(a(n), n=0..30); # Alois P. Heinz, Apr 01 2009
# Computes n -> [a[0],a[1],..,a[n]]
A001003_list := proc(n) local j,a,w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+2*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a,list) end: A001003_list(100); # Peter Luschny, May 17 2011
  • Mathematica
    Table[Length[Flatten[Nest[ #/.a_Integer:> Join[Range[0, a + 1], Range[a, 0, -1]] &, {0}, n]]], {n, 0, 10}]; Sch[ 0 ] = Sch[ 1 ] = 1; Sch[ n_Integer ] := Sch[ n ] = (3(2n - 1)Sch[ n - 1 ] - (n - 2)Sch[ n - 2 ])/(n + 1); Array[ Sch, 24, 0]
(* Second program: *)
a[n_] := Hypergeometric2F1[-n + 1, n + 2, 2, -1]; a[0] = 1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 09 2011, after Vladeta Jovovic *)
a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 6 x + x^2]) / (4 x), {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
Table[(KroneckerDelta[n] - GegenbauerC[n+1, -1/2, 3])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)
a[n_] := -LegendreP[n, -1, 2, 3] I / Sqrt[2]; a[0] = 1;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 16 2019 *)
a[1]:=1; a[2]:=1; a[n_]:=a[n] = a[n-1]+2 Sum[a[k] a[n-k], {k,2,n-1}]; Map[a, Range[24]] (* Oliver Seipel, Nov 03 2024, after Schröder 1870 *)
CoefficientList[InverseSeries[Series[x/(Series[(1 - x)/(1 - 2  x), {x, 0, 24}]), {x, 0, 24}]]/x, x] (* Mats Granvik, Jun 30 2025 *)
  • PARI
    {a(n) = if( n<1, n==0, sum( k=0, n, 2^k * binomial(n, k) * binomial(n, k-1) ) / (2*n))}; /* Michael Somos, Mar 31 2007 */
  • PARI
    {a(n) = my(A); if( n<1, n==0, n--; A = x * O(x^n); n! * simplify( polcoef( exp(3*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* Michael Somos, Mar 31 2007 */
  • PARI
    {a(n) = if( n<0, 0, n++; polcoef( serreverse( (x - 2*x^2) / (1 - x) + x * O(x^n)), n))}; /* Michael Somos, Mar 31 2007 */
  • PARI
    N=30; x='x+O('x^N); Vec( (1+x-(1-6*x+x^2)^(1/2))/(4*x) ) \\ Hugo Pfoertner, Nov 19 2018
  • Python
    # The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)]
def A001003_list(n):
    a = [0 for i in range(n)]
    a[0] = 1
    for w in range(1, n):
        s = 0
        for j in range(1, w):
            s += a[j]*a[w-j-1]
        a[w] = a[w-1]+2*s
    return a
# Peter Luschny, May 17 2011
  • Python
    from gmpy2 import divexact
A001003 = [1, 1]
for n in range(3,10**3):
    A001003.append(divexact(A001003[-1]*(6*n-9)-(n-3)*A001003[-2],n))
# Chai Wah Wu, Sep 01 2014
  • Sage
    # Generalized algorithm of L. Seidel
def A001003_list(n) :
    D = [0]*(n+1); D[1] = 1/2
    b = True; h = 2; R = [1]
    for i in range(2*n-2) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1;
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-1]);
        b = not b
    return R
A001003_list(24) # Peter Luschny, Jun 02 2012

Formula

D-finite with recurrence: (n+1) * a(n) = (6*n-3) * a(n-1) - (n-2) * a(n-2) if n>1. a(0) = a(1) = 1.
a(n) = 3*a(n-1) + 2*A065096(n-2) (n>2). If g(x) = 1 + 3*x + 11*x^2 + 45*x^3 + ... + a(n)*x^n + ..., then g(x) = 1 + 3(x*g(x)) + 2(x*g(x))^2, g(x)^2 = 1 + 6*x + 31*x^2 + 156*x^3 + ... + A065096(n)*x^n + ... - Paul D. Hanna, Jun 10 2002
a(n+1) = -a(n) + 2*Sum_{k=1..n} a(k)*a(n+1-k). - Philippe Deléham, Jan 27 2004
a(n-2) = (1/(n-1))*Sum_{k=0..n-3} binomial(n-1,k+1)*binomial(n-3,k)*2^(n-3-k) for n >= 3 [G. Polya, Elemente de Math., 12 (1957), p. 115.] - N. J. A. Sloane, Jun 13 2015
G.f.: (1 + x - sqrt(1 - 6*x + x^2) )/(4*x) = 2/(1 + x + sqrt(1 - 6*x + x^2)).
a(n) ~ W*(3+sqrt(8))^n*n^(-3/2) where W = (1/4)*sqrt((sqrt(18)-4)/Pi) [See Knuth I, p. 534, or Perez. Note that the formula on line 3, page 475 of Flajolet and Sedgewick seems to be wrong - it has to be multiplied by 2^(1/4).] - N. J. A. Sloane, Apr 10 2011
The correct asymptotic for this sequence is a(n) ~ W*(3+sqrt(8))^n*n^(-3/2), where W = (1+sqrt(2))/(2*2^(3/4)*sqrt(Pi)) = 0.404947065905750651736243... Result in book by D. Knuth (p. 539 of 3rd edition, exercise 12) is for sequence b(n), but a(n) = b(n+1)/2. Therefore is asymptotic a(n) ~ b(n) * (3+sqrt(8))/2. - Vaclav Kotesovec, Sep 09 2012
The Hankel transform of this sequence gives A006125 = 1, 1, 2, 8, 64, 1024, ...; example: det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64. - Philippe Deléham, Mar 02 2004
a(n+1) = Sum_{k=0..floor((n-1)/2)} 2^k * 3^(n-1-2k) * binomial(n-1, 2k) * Catalan(k). This formula counts colored Dyck paths by the same parameter by which Touchard's identity counts ordinary Dyck paths: number of DDUs (U=up step, D=down step). See also Gouyou-Beauchamps reference. - David Callan, Mar 14 2004
From Paul Barry, May 24 2005: (Start)
a(n) = (1/(n+1))*Sum_{k=0..n} C(n+1, k)*C(2n-k, n)*(-1)^k*2^(n-k) [with offset 0].
a(n) = (1/(n+1))*Sum_{k=0..n} C(n+1, k+1)*C(n+k, k)*(-1)^(n-k)*2^k [with offset 0].
a(n) = Sum_{k=0..n} (1/(k+1))*C(n, k)*C(n+k, k)*(-1)^(n-k)*2^k [with offset 0].
a(n) = Sum_{k=0..n} A088617(n, k)*(-1)^(n-k)*2^k [with offset 0]. (End)
E.g.f. of a(n+1) is exp(3*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic, Mar 31 2004
Reversion of (x-2*x^2)/(1-x) is g.f. offset 1.
For n>=1, a(n) = Sum_{k=0..n} 2^k*N(n, k) where N(n, k) = (1/n)*C(n, k)*C(n, k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 10 2003 [This formula counts colored Dyck paths by number of peaks, which is easy to see because the Narayana numbers count Dyck paths by number of peaks and the number of peaks determines the number of interior ascent vertices.]
a(n) = Sum_{k=0..n} A088617(n, k)*2^k*(-1)^(n-k). - Philippe Deléham, Jan 21 2004
For n > 0, a(n) = (1/(n+1)) * Sum_{k = 0 .. n-1} binomial(2*n-k, n) * binomial(n-1, k). This formula counts colored Dyck paths (as above) by number of white vertices. - David Callan, Mar 14 2004
a(n-1) = (d^(n-1)/dx^(n-1))((1-x)/(1-2*x))^n/n!|_{x=0}. (For a proof see the comment on the unsigned row sums of triangle A111785.)
From Wolfdieter Lang, Sep 12 2005: (Start)
a(n) = (1/n)*Sum_{k=1..n} binomial(n, k)*binomial(n+k, k-1).
a(n) = hypergeom([1-n, n+2], [2], -1), n>=1. (End)
a(n) = hypergeom([1-n, -n], [2], 2) for n>=0. - Peter Luschny, Sep 22 2014
a(m+n+1) = Sum_{k>=0} A110440(m, k)*A110440(n, k)*2^k = A110440(m+n, 0). - Philippe Deléham, Sep 14 2005
Sum over partitions formula (reference Schroeder paper p. 362, eq. (1) II). Number the partitions of n according to Abramowitz-Stegun pp. 831-832 (see reference under A105805) with k=1..p(n)= A000041(n). For n>=1: a(n-1) = Sum_{k=2..p(n)} A048996(n,k)*a(1)^e(k, 1)*a(1)^e(k, 2)*...*a(n-2)^e(k, n-1) if the k-th partition of n in the mentioned order is written as (1^e(k, 1), 2^e(k, 2), ..., (n-1)e(k, n-1)). Note that the first (k=1) partition (n^1) has to be omitted. - Wolfdieter Lang, Aug 23 2005
Starting (1, 3, 11, 45, ...), = row sums of triangle A126216 = A001263 * [1, 2, 4, 8, 16, ...]. - Gary W. Adamson, Nov 30 2007
From Paul Barry, May 15 2009: (Start)
G.f.: 1/(1+x-2x/(1+x-2x/(1+x-2x/(1+x-2x/(1-.... (continued fraction).
G.f.: 1/(1-x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction).
G.f.: 1/(1-x-2x^2/(1-3x-2x^2/(1-3x-2x^2/(1-... (continued fraction). (End)
G.f.: 1 / (1 - x / (1 - 2*x / (1 - x / (1 - 2*x / ... )))). - Michael Somos, May 19 2013
a(n) = (LegendreP(n+1,3)-3*LegendreP(n,3))/(4*n) for n>0. - Mark van Hoeij, Jul 12 2010 [This formula is mentioned in S.-J. Kettle's 1982 letter - see link. N. J. A. Sloane, Jun 13 2015]
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, where M is the production matrix:
1, 1, 0, 0, 0, 0, ...
2, 2, 2, 0, 0, 0, ...
1, 1, 1, 1, 0, 0, ...
2, 2, 2, 2, 2, 0, ...
1, 1, 1, 1, 1, 1, ...
... (End)
From Gary W. Adamson, Aug 23 2011: (Start)
a(n) is the sum of top row terms of Q^(n-1), where Q is the infinite square production matrix:
1, 2, 0, 0, 0, ...
1, 1, 2, 0, 0, ...
1, 1, 1, 2, 0, ...
1, 1, 1, 1, 2, ...
... (End)
Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t+3*t^2+4*t^3+...)), an o.g.f. for A003480, then for A001003 a(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. (Cf. A086810.) - Tom Copeland, Sep 06 2011
A006318(n) = 2*a(n) if n>0. - Michael Somos, Mar 31 2007
BINOMIAL transform is A118376 with offset 0. REVERT transform is A153881. INVERT transform is A006318. INVERT transform of A114710. HANKEL transform is A139685. PSUM transform is A104858. - Michael Somos, May 19 2013
G.f.: 1 + x/(Q(0) - x) where Q(k) = 1 + k*(1-x) - x - x*(k+1)*(k+2)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) = A144944(n,n) = A186826(n,0). - Reinhard Zumkeller, May 11 2013
a(n)=(-1)^n*LegendreP(n,-1,-3)/sqrt(2), n > 0, LegendreP(n,a,b) is the Legendre function. - Karol A. Penson, Jul 06 2013
Integral representation as n-th moment of a positive weight function W(x) = W_a(x) + W_c(x), where W_a(x) = Dirac(x)/2, is the discrete (atomic) part, and W_c(x) = sqrt(8-(x-3)^2)/(4*Pi*x) is the continuous part of W(x) defined on (3 sqrt(8),3+sqrt(8)): a(n) = int( x^n*W_a(x), x=-eps..eps ) + int( x^n*W_c(x), x = 3-sqrt(8)..3+sqrt(8) ), for any eps>0, n>=0. W_c(x) is unimodal, of bounded variation and W_c(3-sqrt(8)) = W_c(3+sqrt(8)) = 0. Note that the position of the Dirac peak (x=0) lies outside support of W_c(x). - Karol A. Penson and Wojciech Mlotkowski, Aug 05 2013
G.f.: 1 + x/G(x) with G(x) = 1 - 3*x - 2*x^2/G(x) (continued fraction). - Nikolaos Pantelidis, Dec 17 2022

A074909 Running sum of Pascal's triangle (A007318), or beheaded Pascal's triangle read by beheaded rows.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7, 1, 8, 28, 56, 70, 56, 28, 8, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11
Offset: 0

Views

Author

Wouter Meeussen, Oct 01 2002

Keywords

Comments

This sequence counts the "almost triangular" partitions of n. A partition is triangular if it is of the form 0+1+2+...+k. Examples: 3=0+1+2, 6=0+1+2+3. An "almost triangular" partition is a triangular partition with at most 1 added to each of the parts. Examples: 7 = 1+1+2+3 = 0+2+2+3 = 0+1+3+3 = 0+1+2+4. Thus a(7)=4. 8 = 1+2+2+3 = 1+1+3+3 = 1+1+2+4 = 0+2+3+3 = 0+2+2+4 = 0+1+3+4 so a(8)=6. - Moshe Shmuel Newman, Dec 19 2002
The "almost triangular" partitions are the ones cycled by the operation of "Bulgarian solitaire", as defined by Martin Gardner.
Start with A007318 - I (I = Identity matrix), then delete right border of zeros. - Gary W. Adamson, Jun 15 2007
Also the number of increasing acyclic functions from {1..n-k+1} to {1..n+2}. A function f is acyclic if for every subset B of the domain the image of B under f does not equal B. For example, T(3,1)=4 since there are exactly 4 increasing acyclic functions from {1,2,3} to {1,2,3,4,5}: f1={(1,2),(2,3),(3,4)}, f2={(1,2),(2,3),(3,5)}, f3={(1,2),(2,4),(3,5)} and f4={(1,3),(2,4),(4,5)}. - Dennis P. Walsh, Mar 14 2008
Second Bernoulli polynomials are (from A164555 instead of A027641) B2(n,x) = 1; 1/2, 1; 1/6, 1, 1; 0, 1/2, 3/2, 1; -1/30, 0, 1, 2, 1; 0, -1/6, 0, 5/3, 5/2, 1; ... . Then (B2(n,x)/A002260) = 1; 1/2, 1/2; 1/6, 1/2, 1/3; 0, 1/4, 1/2, 1/4; -1/30, 0, 1/3, 1/2, 1/5; 0, -1/12, 0, 5/12, 1/2, 1/6; ... . See (from Faulhaber 1631) Jacob Bernoulli Summae Potestatum (sum of powers) in A159688. Inverse polynomials are 1; -1, 2; 1, -3, 3; -1, 4, -6, 4; ... = A074909 with negative even diagonals. Reflected A053382/A053383 = reflected B(n,x) = RB(n,x) = 1; -1/2, 1; 1/6, -1, 1; 0, 1/2, -3/2, 1; ... . A074909 is inverse of RB(n,x)/A002260 = 1; -1/2, 1/2; 1/6, -1/2, 1/3; 0, 1/4, -1/2, 1/4; ... . - Paul Curtz, Jun 21 2010
A054143 is the fission of the polynomial sequence (p(n,x)) given by p(n,x) = x^n + x^(n-1) + ... + x + 1 by the polynomial sequence ((x+1)^n). See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011
Reversal of A135278. - Philippe Deléham, Feb 11 2012
For a closed-form formula for arbitrary left and right borders of Pascal-like triangles see A228196. - Boris Putievskiy, Aug 19 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
From A238363, the operator equation d/d(:xD:)f(xD)={exp[d/d(xD)]-1}f(xD) = f(xD+1)-f(xD) follows. Choosing f(x) = x^n and using :xD:^n/n! = binomial(xD,n) and (xD)^n = Bell(n,:xD:), the Bell polynomials of A008277, it follows that the lower triangular matrix [padded A074909]
A) = [St2]*[dP]*[St1] = A048993*A132440*[padded A008275]
B) = [St2]*[dP]*[St2]^(-1)
C) = [St1]^(-1)*[dP]*[St1],
where [St1]=padded A008275 just as [St2]=A048993=padded A008277 whereas [padded A074909]=A007318-I with I=identity matrix. - Tom Copeland, Apr 25 2014
T(n,k) generated by m-gon expansions in the case of odd m with "vertex to side" version or even m with "vertex to vertes" version. Refer to triangle expansions in A061777 and A101946 (and their companions for m-gons) which are "vertex to vertex" and "vertex to side" versions respectively. The label values at each iteration can be arranged as a triangle. Any m-gon can also be arranged as the same triangle with conditions: (i) m is odd and expansion is "vertex to side" version or (ii) m is even and expansion is "vertex to vertex" version. m*Sum_{i=1..k} T(n,k) gives the total label value at the n-th iteration. See also A247976. Vertex to vertex: A061777, A247618, A247619, A247620. Vertex to side: A101946, A247903, A247904, A247905. - Kival Ngaokrajang Sep 28 2014
From Tom Copeland, Nov 12 2014: (Start)
With P(n,x) = [(x+1)^(n+1)-x^(n+1)], the row polynomials of this entry, Up(n,x) = P(n,x)/(n+1) form an Appell sequence of polynomials that are the umbral compositional inverses of the Bernoulli polynomials B(n,x), i.e., B[n,Up(.,x)] = x^n = Up[n,B(.,x)] under umbral substitution, e.g., B(.,x)^n = B(n,x).
The e.g.f. for the Bernoulli polynomials is [t/(e^t - 1)] e^(x*t), and for Up(n,x) it's exp[Up(.,x)t] = [(e^t - 1)/t] e^(x*t).
Another g.f. is G(t,x) = log[(1-x*t)/(1-(1+x)*t)] = log[1 + t /(1 + -(1+x)t)] = t/(1-t*Up(.,x)) = Up(0,x)*t + Up(1,x)*t^2 + Up(2,x)*t^3 + ... = t + (1+2x)/2 t^2 + (1+3x+3x^2)/3 t^3 + (1+4x+6x^2+4x^3)/4 t^4 + ... = -log(1-t*P(.,x)), expressed umbrally.
The inverse, Ginv(t,x), in t of the g.f. may be found in A008292 from Copeland's list of formulas (Sep 2014) with a=(1+x) and b=x. This relates these two sets of polynomials to algebraic geometry, e.g., elliptic curves, trigonometric expansions, Chebyshev polynomials, and the combinatorics of permutahedra and their duals.
Ginv(t,x) = [e^((1+x)t) - e^(xt)] / [(1+x) * e^((1+x)t) - x * e^(xt)] = [e^(t/2) - e^(-t/2)] / [(1+x)e^(t/2) - x*e^(-t/2)] = (e^t - 1) / [1 + (1+x) (e^t - 1)] = t - (1 + 2 x) t^2/2! + (1 + 6 x + 6 x^2) t^3/3! - (1 + 14 x + 36 x^2 + 24 x^3) t^4/4! + ... = -exp[-Perm(.,x)t], where Perm(n,x) are the reverse face polynomials, or reverse f-vectors, for the permutahedra, i.e., the face polynomials for the duals of the permutahedra. Cf. A090582, A019538, A049019, A133314, A135278.
With L(t,x) = t/(1+t*x) with inverse L(t,-x) in t, and Cinv(t) = e^t - 1 with inverse C(t) = log(1 + t). Then Ginv(t,x) = L[Cinv(t),(1+x)] and G(t,x) = C[L[t,-(1+x)]]. Note L is the special linear fractional (Mobius) transformation.
Connections among the combinatorics of the permutahedra, simplices (cf. A135278), and the associahedra can be made through the Lagrange inversion formula (LIF) of A133437 applied to G(t,x) (cf. A111785 and the Schroeder paths A126216 also), and similarly for the LIF A134685 applied to Ginv(t,x) involving the simplicial Whitehouse complex, phylogenetic trees, and other structures. (See also the LIFs A145271 and A133932). (End)
R = x - exp[-[B(n+1)/(n+1)]D] = x - exp[zeta(-n)D] is the raising operator for this normalized sequence UP(n,x) = P(n,x) / (n+1), that is, R UP(n,x) = UP(n+1,x), where D = d/dx, zeta(-n) is the value of the Riemann zeta function evaluated at -n, and B(n) is the n-th Bernoulli number, or constant B(n,0) of the Bernoulli polynomials. The raising operator for the Bernoulli polynomials is then x + exp[-[B(n+1)/(n+1)]D]. [Note added Nov 25 2014: exp[zeta(-n)D] is abbreviation of exp(a.D) with (a.)^n = a_n = zeta(-n)]. - Tom Copeland, Nov 17 2014
The diagonals T(n, n-m), for n >= m, give the m-th iterated partial sum of the positive integers; that is A000027(n+1), A000217(n), A000292(n-1), A000332(n+1), A000389(n+1), A000579(n+1), A000580(n+1), A000581(n+1), A000582(n+1), ... . - Wolfdieter Lang, May 21 2015
The transpose gives the numerical coefficients of the Maurer-Cartan form matrix for the general linear group GL(n,1) (cf. Olver, but note that the formula at the bottom of p. 6 has an error--the 12 should be a 15). - Tom Copeland, Nov 05 2015
The left invariant Maurer-Cartan form polynomial on p. 7 of the Olver paper for the group GL^n(1) is essentially a binomial convolution of the row polynomials of this entry with those of A133314, or equivalently the row polynomials generated by the product of the e.g.f. of this entry with that of A133314, with some reindexing. - Tom Copeland, Jul 03 2018
From Tom Copeland, Jul 10 2018: (Start)
The first column of the inverse matrix is the sequence of Bernoulli numbers, which follows from the umbral definition of the Bernoulli polynomials (B.(0) + x)^n = B_n(x) evaluated at x = 1 and the relation B_n(0) = B_n(1) for n > 1 and -B_1(0) = 1/2 = B_1(1), so the Bernoulli numbers can be calculated using Cramer's rule acting on this entry's matrix and, therefore, from the ratios of volumes of parallelepipeds determined by the columns of this entry's square submatrices. - Tom Copeland, Jul 10 2018
Umbrally composing the row polynomials with B_n(x), the Bernoulli polynomials, gives (B.(x)+1)^(n+1) - (B.(x))^(n+1) = d[x^(n+1)]/dx = (n+1)*x^n, so multiplying this entry as a lower triangular matrix (LTM) by the LTM of the coefficients of the Bernoulli polynomials gives the diagonal matrix of the natural numbers. Then the inverse matrix of this entry has the elements B_(n,k)/(k+1), where B_(n,k) is the coefficient of x^k for B_n(x), and the e.g.f. (1/x) (e^(xt)-1)/(e^t-1). (End)

Examples

			T(4,2) = 0+0+1+3+6 = 10 = binomial(5, 2).
Triangle T(n,k) begins:
n\k 0  1  2   3   4   5   6   7   8   9 10 11
0:  1
1:  1  2
2:  1  3  3
3:  1  4  6   4
4:  1  5 10  10   5
5:  1  6 15  20  15   6
6:  1  7 21  35  35  21   7
7:  1  8 28  56  70  56  28   8
8:  1  9 36  84 126 126  84  36  9
9:  1 10 45 120 210 252 210 120 45   10
10: 1 11 55 165 330 462 462 330 165  55 11
11: 1 12 66 220 495 792 924 792 495 220 66 12
... Reformatted. - _Wolfdieter Lang_, Nov 04 2014
.
Can be seen as the square array A(n, k) = binomial(n + k + 1, n) read by descending antidiagonals. A(n, k) is the number of monotone nondecreasing functions f: {1,2,..,k} -> {1,2,..,n}. - _Peter Luschny_, Aug 25 2019
[0]  1,  1,   1,   1,    1,    1,     1,     1,     1, ... A000012
[1]  2,  3,   4,   5,    6,    7,     8,     9,    10, ... A000027
[2]  3,  6,  10,  15,   21,   28,    36,    45,    55, ... A000217
[3]  4, 10,  20,  35,   56,   84,   120,   165,   220, ... A000292
[4]  5, 15,  35,  70,  126,  210,   330,   495,   715, ... A000332
[5]  6, 21,  56, 126,  252,  462,   792,  1287,  2002, ... A000389
[6]  7, 28,  84, 210,  462,  924,  1716,  3003,  5005, ... A000579
[7]  8, 36, 120, 330,  792, 1716,  3432,  6435, 11440, ... A000580
[8]  9, 45, 165, 495, 1287, 3003,  6435, 12870, 24310, ... A000581
[9] 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, ... A000582

Links

Crossrefs

Cf. A007318, A181971, A228196, A228576.
Row sums are A000225, diagonal sums are A052952.
The number of acyclic functions is A058127.
Cf. A008292, A090582, A019538, A049019, A133314, A135278, A133437, A111785, A126216, A134685, A133932, A248727, A033282, A134264.
Cf. A000027, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582.
Cf. A325191.

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->Binomial(n+1,k)))); # Muniru A Asiru, Jul 10 2018
  • Haskell
    a074909 n k = a074909_tabl !! n !! k
a074909_row n = a074909_tabl !! n
a074909_tabl = iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [1])) [1]
-- Reinhard Zumkeller, Feb 25 2012
  • Magma
    /* As triangle */ [[Binomial(n+1,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 22 2018
  • Maple
    A074909 := proc(n,k)
    if k > n or k < 0 then
        0;
    else
        binomial(n+1,k) ;
    end if;
end proc: # Zerinvary Lajos, Nov 09 2006
  • Mathematica
    Flatten[Join[{1}, Table[Sum[Binomial[k, m], {k, 0, n}], {n, 0, 12}, {m, 0, n}] ]] (* or *) Flatten[Join[{1}, Table[Binomial[n, m], {n, 12}, {m, n}]]]
  • PARI
    print1(1);for(n=1,10,for(k=1,n,print1(", "binomial(n,k)))) \\ Charles R Greathouse IV, Mar 26 2013
  • Python
    from math import comb, isqrt
def A074909(n): return comb(r:=(m:=isqrt(k:=n+1<<1))+(k>m*(m+1)),n-comb(r,2)) # Chai Wah Wu, Nov 12 2024

Formula

T(n, k) = Sum_{i=0..n} C(i, n-k) = C(n+1, k).
Row n has g.f. (1+x)^(n+1)-x^(n+1).
E.g.f.: ((1+x)*e^t - x) e^(x*t). The row polynomials p_n(x) satisfy dp_n(x)/dx = (n+1)*p_(n-1)(x). - Tom Copeland, Jul 10 2018
T(n, k) = T(n-1, k-1) + T(n-1, k) for k: 0Reinhard Zumkeller, Apr 18 2005
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(1,0)=1, T(1,1)=2, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 27 2013
G.f. for column k (with leading zeros): x^(k-1)*(1/(1-x)^(k+1)-1), k >= 0. - Wolfdieter Lang, Nov 04 2014
Up(n, x+y) = (Up(.,x)+ y)^n = Sum_{k=0..n} binomial(n,k) Up(k,x)*y^(n-k), where Up(n,x) = ((x+1)^(n+1)-x^(n+1)) / (n+1) = P(n,x)/(n+1) with P(n,x) the n-th row polynomial of this entry. dUp(n,x)/dx = n * Up(n-1,x) and dP(n,x)/dx = (n+1)*P(n-1,x). - Tom Copeland, Nov 14 2014
The o.g.f. GF(x,t) = x / ((1-t*x)*(1-(1+t)x)) = x + (1+2t)*x^2 + (1+3t+3t^2)*x^3 + ... has the inverse GFinv(x,t) = (1+(1+2t)x-sqrt(1+(1+2t)*2x+x^2))/(2t(1+t)x) in x about 0, which generates the row polynomials (mod row signs) of A033282. The reciprocal of the o.g.f., i.e., x/GF(x,t), gives the free cumulants (1, -(1+2t) , t(1+t) , 0, 0, ...) associated with the moments defined by GFinv, and, in fact, these free cumulants generate these moments through the noncrossing partitions of A134264. The associated e.g.f. and relations to Grassmannians are described in A248727, whose polynomials are the basis for an Appell sequence of polynomials that are umbral compositional inverses of the Appell sequence formed from this entry's polynomials (distinct from the one described in the comments above, without the normalizing reciprocal). - Tom Copeland, Jan 07 2015
T(n, k) = (1/k!) * Sum_{i=0..k} Stirling1(k,i)*(n+1)^i, for 0<=k<=n. - Ridouane Oudra, Oct 23 2022

Extensions

I added an initial 1 at the suggestion of Paul Barry, which makes the triangle a little nicer but may mean that some of the formulas will now need adjusting. - N. J. A. Sloane, Feb 11 2003
Formula section edited, checked and corrected by Wolfdieter Lang, Nov 04 2014

A126120 Catalan numbers (A000108) interpolated with 0's.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0
Offset: 0

Views

Author

Philippe Deléham, Mar 06 2007

Keywords

Comments

Inverse binomial transform of A001006.
The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].
Counts returning walks (excursions) of length n on a 1-d integer lattice with step set {+1,-1} which stay in the chamber x >= 0. - Andrew V. Sutherland, Feb 29 2008
Moment sequence of the trace of a random matrix in G=USp(2)=SU(2). If X=tr(A) is a random variable (A distributed according to the Haar measure on G) then a(n) = E[X^n]. - Andrew V. Sutherland, Feb 29 2008
Essentially the same as A097331. - R. J. Mathar, Jun 15 2008
Number of distinct proper binary trees with n nodes. - Chris R. Sims (chris.r.sims(AT)gmail.com), Jun 30 2010
-a(n-1), with a(-1):=0, n>=0, is the Z-sequence for the Riordan array A049310 (Chebyshev S). For the definition see that triangle. - Wolfdieter Lang, Nov 04 2011
See A180874 (also A238390 and A097610) and A263916 for relations to the general Bell A036040, cycle index A036039, and cumulant expansion polynomials A127671 through the Faber polynomials. - Tom Copeland, Jan 26 2016
A signed version is generated by evaluating polynomials in A126216 that are essentially the face polynomials of the associahedra. This entry's sequence is related to an inversion relation on p. 34 of Mizera, related to Feynman diagrams. - Tom Copeland, Dec 09 2019

Examples

			G.f. = 1 + x^2 + 2*x^4 + 5*x^6 + 14*x^8 + 42*x^10 + 132*x^12 + 429*x^14 + ...
From _Gus Wiseman_, Nov 14 2022: (Start)
The a(0) = 1 through a(8) = 14 ordered binary rooted trees with n + 1 nodes (ranked by A358375):
  o  .  (oo)  .  ((oo)o)  .  (((oo)o)o)  .  ((((oo)o)o)o)
                 (o(oo))     ((o(oo))o)     (((o(oo))o)o)
                             ((oo)(oo))     (((oo)(oo))o)
                             (o((oo)o))     (((oo)o)(oo))
                             (o(o(oo)))     ((o((oo)o))o)
                                            ((o(o(oo)))o)
                                            ((o(oo))(oo))
                                            ((oo)((oo)o))
                                            ((oo)(o(oo)))
                                            (o(((oo)o)o))
                                            (o((o(oo))o))
                                            (o((oo)(oo)))
                                            (o(o((oo)o)))
                                            (o(o(o(oo))))
(End)

References

  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.

Links

Crossrefs

Cf. A000108, A036039, A036040, A097610, A127671, A180874, A238390, A263916.
Cf. A126216.
The unordered version is A001190, ranked by A111299.
These trees (ordered binary rooted) are ranked by A358375.
Cf. A000081, A001263, A005043, A032027, A063895, A245824.

Programs

  • Magma
    &cat [[Catalan(n), 0]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2016
  • Maple
    with(combstruct): grammar := { BB = Sequence(Prod(a,BB,b)), a = Atom, b = Atom }: seq(count([BB,grammar], size=n),n=0..47); # Zerinvary Lajos, Apr 25 2007
BB := {E=Prod(Z,Z), S=Union(Epsilon,Prod(S,S,E))}: ZL:=[S,BB,unlabeled]: seq(count(ZL, size=n), n=0..45); # Zerinvary Lajos, Apr 22 2007
BB := [T,{T=Prod(Z,Z,Z,F,F), F=Sequence(B), B=Prod(F,Z,Z)}, unlabeled]: seq(count(BB, size=n+1), n=0..45); # valid for n> 0. # Zerinvary Lajos, Apr 22 2007
seq(n!*coeff(series(hypergeom([],[2],x^2),x,n+2),x,n),n=0..45); # Peter Luschny, Jan 31 2015
# Using function CompInv from A357588.
CompInv(48, n -> ifelse(irem(n, 2) = 0, 0, (-1)^iquo(n-1, 2))); # Peter Luschny, Oct 07 2022
  • Mathematica
    a[n_?EvenQ] := CatalanNumber[n/2]; a[n_] = 0; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Sep 10 2012 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ BesselI[ 1, 2 x] / x, {x, 0, n}]]; (* Michael Somos, Mar 19 2014 *)
bot[n_]:=If[n==1,{{}},Join@@Table[Tuples[bot/@c],{c,Table[{k,n-k-1},{k,n-1}]}]];
Table[Length[bot[n]],{n,10}] (* Gus Wiseman, Nov 14 2022 *)
Riffle[CatalanNumber[Range[0,50]],0,{2,-1,2}] (* Harvey P. Dale, May 28 2024 *)
  • Python
    from math import comb
def A126120(n): return 0 if n&1 else comb(n,m:=n>>1)//(m+1) # Chai Wah Wu, Apr 22 2024
  • Sage
    def A126120_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 2; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] -= D[k-1]
            h += 1; R.append(abs(D[1]))
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
    return R
A126120_list(46) # Peter Luschny, Jun 03 2012

Formula

a(2*n) = A000108(n), a(2*n+1) = 0.
a(n) = A053121(n,0).
(1/Pi) Integral_{0 .. Pi} (2*cos(x))^n *2*sin^2(x) dx. - Andrew V. Sutherland, Feb 29 2008
G.f.: (1 - sqrt(1 - 4*x^2)) / (2*x^2) = 1/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). - Philippe Deléham, Nov 24 2009
G.f. A(x) satisfies A(x) = 1 + x^2*A(x)^2. - Vladimir Kruchinin, Feb 18 2011
E.g.f.: I_1(2x)/x Where I_n(x) is the modified Bessel function. - Benjamin Phillabaum, Mar 07 2011
Apart from the first term the e.g.f. is given by x*HyperGeom([1/2],[3/2,2], x^2). - Benjamin Phillabaum, Mar 07 2011
a(n) = Integral_{x=-2..2} x^n*sqrt((2-x)*(2+x))/(2*Pi) dx. - Peter Luschny, Sep 11 2011
E.g.f.: E(0)/(1-x) where E(k) = 1-x/(1-x/(x-(k+1)*(k+2)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
G.f.: 3/2- sqrt(1-4*x^2)/2 = 1/x^2 + R(0)/x^2, where R(k) = 2*k-1 - x^2*(2*k-1)*(2*k+1)/R(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013 (warning: this is not the g.f. of this sequence, R. J. Mathar, Sep 23 2021)
G.f.: 1/Q(0), where Q(k) = 2*k+1 + x^2*(1-4*(k+1)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014
a(n) = n!*[x^n]hypergeom([],[2],x^2). - Peter Luschny, Jan 31 2015
a(n) = 2^n*hypergeom([3/2,-n],[3],2). - Peter Luschny, Feb 03 2015
a(n) = ((-1)^n+1)*2^(2*floor(n/2)-1)*Gamma(floor(n/2)+1/2)/(sqrt(Pi)* Gamma(floor(n/2)+2)). - Ilya Gutkovskiy, Jul 23 2016
D-finite with recurrence (n+2)*a(n) +4*(-n+1)*a(n-2)=0. - R. J. Mathar, Mar 21 2021
From Peter Bala, Feb 03 2024: (Start)
a(n) = 2^n * Sum_{k = 0..n} (-2)^(-k)*binomial(n, k)*Catalan(k+1).
G.f.: 1/(1 + 2*x) * c(x/(1 + 2*x))^2 = 1/(1 - 2*x) * c(-x/(1 - 2*x))^2 = c(x^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

Extensions

An erroneous comment removed by Tom Copeland, Jul 23 2016

A033282 Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 1, 9, 21, 14, 1, 14, 56, 84, 42, 1, 20, 120, 300, 330, 132, 1, 27, 225, 825, 1485, 1287, 429, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796
Offset: 3

Views

Author

N. J. A. Sloane

Keywords

Comments

T(n+3, k) is also the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of the triangle of Narayana numbers A001263. For example, x^2 + 5*x + 5 = y^2 + 3*y + 1. - Paul Boddington, Mar 07 2003
Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 1's (see the Stanley reference).
Number of k-dimensional 'faces' of the n-dimensional associahedron (see Simion, p. 168). - Mitch Harris, Jan 16 2007
Mirror image of triangle A126216. - Philippe Deléham, Oct 19 2007
For relation to Lagrange inversion or series reversion and the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. - Tom Copeland, Sep 29 2008
Row generating polynomials 1/(n+1)*Jacobi_P(n,1,1,2*x+1). Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p. 60]. See A001263 for the corresponding array of h-vectors for associahedra of type A_n. See A063007 and A080721 for the f-vectors for associahedra of type B and type D respectively. - Peter Bala, Oct 28 2008
f-vectors of secondary polytopes for Grobner bases for optimization and integer programming (see De Loera et al. and Thomas). - Tom Copeland, Oct 11 2011
From Devadoss and O'Rourke's book: The Fulton-MacPherson compactification of the configuration space of n free particles on a line segment with a fixed particle at each end is the n-Dim Stasheff associahedron whose refined f-vector is given in A133437 which reduces to A033282. - Tom Copeland, Nov 29 2011
Diagonals of A132081 are rows of A033282. - Tom Copeland, May 08 2012
The general results on the convolution of the refined partition polynomials of A133437, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. - Tom Copeland, Sep 20 2016
The signed triangle t(n, k) =(-1)^k* T(n+2, k-1), n >= 1, k = 1..n, seems to be obtainable from the partition array A111785 (in Abramowitz-Stegun order) by adding the entries corresponding to the partitions of n with the number of parts k. E.g., triangle t, row n=4: -1, (6+3) = 9, -21, 14. - Wolfdieter Lang, Mar 17 2017
The preceding conjecture by Lang is true. It is implicit in Copeland's 2011 comments in A086810 on the relations among a gf and its compositional inverse for that entry and inversion through A133437 (a differently normalized version of A111785), whose integer partitions are the same as those for A134685. (An inversion pair in Copeland's 2008 formulas below can also be used to prove the conjecture.) In addition, it follows from the relation between the inversion formula of A111785/A133437 and the enumeration of distinct faces of associahedra. See the MathOverflow link concernimg Loday and the Aguiar and Ardila reference in A133437 for proofs of the relations between the partition polynomials for inversion and enumeration of the distinct faces of the A_n associahedra, or Stasheff polytopes. - Tom Copeland, Dec 21 2017
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)/(n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 07 2022
Chapoton's observation above is correct: the precise expansion is (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)/ (n!*(n+1)!) = Sum_{k = 0..n-1} (-1)^k*T(n+2,n-k-1)*binomial(x+2*n-k,2*n-k), as can be verified using the WZ algorithm. For example, n = 4 gives (x+1)*(x+2)^2*(x+3)^2*(x+4)^2*(x+5)/(4!*5!) = 14*binomial(x+8,8) - 21*binomial(x+7,7) + 9*binomial(x+6,6) - binomial(x+5,5). - Peter Bala, Jun 24 2023

Examples

			The triangle T(n, k) begins:
n\k  0  1   2    3     4     5      6      7     8     9
3:   1
4:   1  2
5:   1  5   5
6:   1  9  21   14
7:   1 14  56   84    42
8:   1 20 120  300   330   132
9:   1 27 225  825  1485  1287    429
10:  1 35 385 1925  5005  7007   5005   1430
11:  1 44 616 4004 14014 28028  32032  19448  4862
12:  1 54 936 7644 34398 91728 148512 143208 75582 16796
... reformatted. - _Wolfdieter Lang_, Mar 17 2017

References

  • S. Devadoss and J. O'Rourke, Discrete and Computational Geometry, Princeton Univ. Press, 2011 (See p. 241.)
  • Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. Exercise 7.50, pages 379, 573.
  • T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 5.8.

Links

Crossrefs

Cf. diagonals: A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281; row sums: A001003 (Schroeder numbers, first term omitted). See A086810 for another version.
A007160 is a diagonal. Cf. A001263.
With leading zero: A086810.
Cf. A019538 'faces' of the permutohedron.
Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image).
Cf. A248727 for a relation to f-polynomials of simplices.
Cf. A111785 (contracted partition array, unsigned; see a comment above).
Antidiagonal sums give A005043. - Jordan Tirrell, Jun 01 2017

Programs

  • Magma
    [[Binomial(n-3, k)*Binomial(n+k-1, k)/(k+1): k in [0..(n-3)]]: n in [3..12]];  // G. C. Greubel, Nov 19 2018
  • Maple
    T:=(n,k)->binomial(n-3,k)*binomial(n+k-1,k)/(k+1): seq(seq(T(n,k),k=0..n-3),n=3..12); # Muniru A Asiru, Nov 24 2018
  • Mathematica
    t[n_, k_] = Binomial[n-3, k]*Binomial[n+k-1, k]/(k+1);
Flatten[Table[t[n, k], {n, 3, 12}, {k, 0, n-3}]][[1 ;; 52]] (* Jean-François Alcover, Jun 16 2011 *)
  • PARI
    Q=(1+z-(1-(4*w+2+O(w^20))*z+z^2+O(z^20))^(1/2))/(2*(1+w)*z);for(n=3,12,for(m=1,n-2,print1(polcoef(polcoef(Q,n-2,z),m,w),", "))) \\ Hugo Pfoertner, Nov 19 2018
  • PARI
    for(n=3,12, for(k=0,n-3, print1(binomial(n-3,k)*binomial(n+k-1,k)/(k+1), ", "))) \\ G. C. Greubel, Nov 19 2018
  • Sage
    [[ binomial(n-3,k)*binomial(n+k-1,k)/(k+1) for k in (0..(n-3))] for n in (3..12)] # G. C. Greubel, Nov 19 2018

Formula

G.f. G = G(t, z) satisfies (1+t)*G^2 - z*(1-z-2*t*z)*G + t*z^4 = 0.
T(n, k) = binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <= k <= n-3.
From Tom Copeland, Nov 03 2008: (Start)
Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.
1. a: f1(x,t) = y = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/[2x (t+1)] = t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...
b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y] - 1/(1+y)} = (y/t) - (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...
2. a: f2(x,t) = y = {1 - x - sqrt[(1-x)^2 - 4xt]}/[2(t+1)] = (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...
b: x2 = y(t+1) [1- y(t+1)]/[t + y(t+1)] = (t+1) (y/t) - (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...
c: y/x2(y,t) = [t/(t+1) + y] / [1- y(t+1)] = t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...
x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437) to generate A033282 and show that A133437 is a refinement of A033282, i.e., a refinement of the f-polynomials of the associahedra, the Stasheff polytopes.
y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264) to generate A033282 and show that A134264 is a refinement of A001263, i.e., a refinement of the h-polynomials of the associahedra.
f1[x,t](t+1) gives a generator for A088617.
f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].
f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].
The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)
G.f.: 1/(1-x*y-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-.... (continued fraction). - Paul Barry, Feb 06 2009
Let h(t) = (1-t)^2/(1+(u-1)*(1-t)^2) = 1/(u + 2*t + 3*t^2 + 4*t^3 + ...), then a signed (n-1)-th row polynomial of A033282 is given by u^(2n-1)*(1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=2. The power series expansion of h(t) is related to A181289 (cf. A086810). - Tom Copeland, Sep 06 2011
With a different offset, the row polynomials equal 1/(1 + x)*Integral_{0..x} R(n,t) dt, where R(n,t) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k,k)*t^k are the row polynomials of A063007. - Peter Bala, Jun 23 2016
n-th row polynomial = ( LegendreP(n-1,2*x + 1) - LegendreP(n-3,2*x + 1) )/((4*n - 6)*x*(x + 1)), n >= 3. - Peter Bala, Feb 22 2017
n*T(n+1, k) = (4n-6)*T(n, k-1) + (2n-3)*T(n, k) - (n-3)*T(n-1, k) for n >= 4. - Fang Lixing, May 07 2019

Extensions

Missing factor of 2 for expansions of f1 and f2 added by Tom Copeland, Apr 12 2009

A060693 Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 14, 35, 30, 10, 1, 42, 126, 140, 70, 15, 1, 132, 462, 630, 420, 140, 21, 1, 429, 1716, 2772, 2310, 1050, 252, 28, 1, 1430, 6435, 12012, 12012, 6930, 2310, 420, 36, 1, 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1, 16796
Offset: 0

Views

Author

F. Chapoton, Apr 20 2001

Keywords

Comments

The rows sum to A006318 (Schroeder numbers), the left column is A000108 (Catalan numbers); the next-to-left column is A001700, the alternating sum in each row but the first is 0.
T(n,k) is the number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. E.g., T(n,k) = binomial(n,k)*binomial(2n-k,n-1)/n for n>0. - Emeric Deutsch, Dec 06 2003
A090181*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 14 2008
T(n,k) is also the number of rooted plane trees with maximal degree 3 and k vertices of degree 2 (a node may have at most 2 children, and there are exactly k nodes with 1 child). Equivalently, T(n,k) is the number of syntactically different expressions that can be formed that use a unary operation k times, a binary operation n-k times, and nothing else (sequence of operands is fixed). - Lars Hellstrom (Lars.Hellstrom(AT)residenset.net), Dec 08 2009

Examples

			Triangle begins:
00: [    1]
01: [    1,     1]
02: [    2,     3,      1]
03: [    5,    10,      6,      1]
04: [   14,    35,     30,     10,      1]
05: [   42,   126,    140,     70,     15,      1]
06: [  132,   462,    630,    420,    140,     21,     1]
07: [  429,  1716,   2772,   2310,   1050,    252,    28,    1]
08: [ 1430,  6435,  12012,  12012,   6930,   2310,   420,   36,   1]
09: [ 4862, 24310,  51480,  60060,  42042,  18018,  4620,  660,  45,  1]
10: [16796, 92378, 218790, 291720, 240240, 126126, 42042, 8580, 990, 55, 1]
...

Links

Crossrefs

Cf. A006318, A000108, A001700.
Triangle in A088617 transposed.
Diagonals give A000108, A001700, A002457, A002802, A002803; A000012, A000217, A034827, A000910, A088625, A088626.
T(2n,n) gives A007004.
Cf. A001263, A033282, A086810, A088617, A097610, A101282.

Programs

  • Maple
    A060693 := (n,k) -> binomial(n,k)*binomial(2*n-k,n)/(n-k+1); # Peter Luschny, May 17 2011
  • Mathematica
    t[n_, k_] := Binomial[n, k]*Binomial[2 n - k, n]/(n - k + 1); Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Robert G. Wilson v, May 30 2011 *)
  • PARI
    T(n, k) = binomial(n, k)*binomial(2*n - k, n)/(n - k + 1);
for(n=0, 10, for(k=0, n, print1(T(n, k),", ")); print); \\ Indranil Ghosh, Jul 28 2017
  • Python
    from sympy import binomial
def T(n, k): return binomial(n, k) * binomial(2 * n - k, n) / (n - k + 1)
for n in range(11): print([T(n, k) for k in range(n + 1)])  # Indranil Ghosh, Jul 28 2017

Formula

Triangle T(n, k) (0 <= k <= n) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 12 2003
If C_n(x) is the g.f. of row n of the Narayana numbers (A001263), C_n(x) = Sum_{k=1..n} binomial(n,k-1)*(binomial(n-1,k-1)/k) * x^k and T_n(x) is the g.f. of row n of T(n,k), then T_n(x) = C_n(x+1), or T(n,k) = [x^n]Sum_{k=1..n}(A001263(n,k)*(x+1)^k). - Mitch Harris, Jan 16 2007, Jan 31 2007
G.f.: (1 - t*y - sqrt((1-y*t)^2 - 4*y)) / 2.
T(n, k) = binomial(2n-k, n)*binomial(n, k)/(n-k+1). - Philippe Deléham, Dec 07 2003
A060693(n, k) = binomial(2*n-k, k)*A000108(n-k); A000108: Catalan numbers. - Philippe Deléham, Dec 30 2003
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Apr 01 2007
T(n,k) = Sum_{j>=0} A090181(n,j)*binomial(j,k). - Philippe Deléham, May 04 2007
Sum_{k=0..n} T(n,k)*x^(n-k) = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Oct 18 2007
From Paul Barry, Jan 29 2009: (Start)
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-.... (continued fraction);
G.f.: 1/(1-(x+xy)/(1-x/(1-(x+xy)/(1-x/(1-(x+xy)/(1-.... (continued fraction). (End)
T(n,k) = [k<=n]*(Sum_{j=0..n} binomial(n,j)^2*binomial(j,k))/(n-k+1). - Paul Barry, May 28 2009
T(n,k) = A104684(n,k)/(n-k+1). - Peter Luschny, May 17 2011
From Tom Copeland, Sep 21 2011: (Start)
With F(x,t) = (1-(2+t)*x-sqrt(1-2*(2+t)*x+(t*x)^2))/(2*x) an o.g.f. (nulling the n=0 term) in x for the A060693 polynomials in t,
G(x,t) = x/(1+t+(2+t)*x+x^2) is the compositional inverse in x.
Consequently, with H(x,t) = 1/(dG(x,t)/dx) = (1+t+(2+t)*x+x^2)^2 / (1+t-x^2), the n-th A060693 polynomial in t is given by (1/n!)*((H(x,t)*d/dx)^n) x evaluated at x=0, i.e., F(x,t) = exp(x*H(u,t)*d/d) u, evaluated at u = 0.
Also, dF(x,t)/dx = H(F(x,t),t). (End)
See my 2008 formulas in A033282 to relate this entry to A088617, A001263, A086810, and other matrices. - Tom Copeland, Jan 22 2016
Rows of this entry are non-vanishing antidiagonals of A097610. See p. 14 of Agapito et al. for a bivariate generating function and its inverse. - Tom Copeland, Feb 03 2016
From Werner Schulte, Jan 09 2017: (Start)
T(n,k) = A126216(n,k-1) + A126216(n,k) for 0 < k < n;
Sum_{k=0..n} (-1)^k*(1+x*(n-k))*T(n,k) = x + (1-x)*A000007(n).
(End)
Conjecture: Sum_{k=0..n} (-1)^k*T(n,k)*(n+1-k)^2 = 1+n+n^2. - Werner Schulte, Jan 11 2017

Extensions

More terms from Vladeta Jovovic, Apr 21 2001
New description from Philippe Deléham, Aug 12 2003
New name using a comment by Emeric Deutsch from Peter Luschny, Jul 26 2017

A133437 Irregular triangle of coefficients of a partition transform for direct Lagrange inversion of an o.g.f., complementary to A134685 for an e.g.f.; normalized by the factorials, these are signed, refined face polynomials of the associahedra.

Original entry on oeis.org

1, -2, 12, -6, -120, 120, -24, 1680, -2520, 360, 720, -120, -30240, 60480, -20160, -20160, 5040, 5040, -720, 665280, -1663200, 907200, 604800, -60480, -362880, -181440, 20160, 40320, 40320, -5040, -17297280, 51891840, -39916800, -19958400, 6652800, 19958400, 6652800, -1814400, -1814400, -3628800, -1814400, 362880, 362880, 362880, -40320
Offset: 1

Views

Author

Tom Copeland, Jan 27 2008

Keywords

Comments

Let f(t) = u(t) - u(0) = Sum_{n>=1} u_n * t^n.
If u_1 is not equal to 0, then the compositional inverse for f(t) is given by g(t) = Sum_{j>=1} P(n,t) where, with u_n denoted by (n'),
P(1,t) = (1')^(-1) * [ 1 ] * t
P(2,t) = (1')^(-3) * [ -2 (2') ] * t^2 / 2!
P(3,t) = (1')^(-5) * [ 12 (2')^2 - 6 (1')(3') ] * t^3 / 3!
P(4,t) = (1')^(-7) * [ -120 (2')^3 + 120 (1')(2')(3') - 24 (1')^2 (4') ] * t^4 / 4!
P(5,t) = (1')^(-9) * [ 1680 (2')^4 - 2520 (1') (2')^2 (3') + 360 (1')^2 (3')^2 + 720 (1')^2 (2') (4') - 120 (1')^3 (5') ] * t^5 / 5!
P(6,t) = (1')^(-11) * [ -30240 (2')^5 + 60480 (1') (2')^3 (3') - 20160 (1')^2 (2') (3')^2 - 20160 (1')^2 (2')^2 (4') + 5040 (1')^3 (3')(4') + 5040 (1')^3 (2')(5') - 720 (1')^4 (6') ] * t^6 / 6!
P(7,t) = (1')^(-13) * [ 665280 (2')^6 - 1663200 (1')(2')^4(3') + (1')^2 [907200 (2')^2(3')^2 + 604800 (2')^3(4')] - (1')^3 [60480 (3')^3 + 362880 (2')(3')(4') + 181440 (2')^2(5')] + (1')^4 [20160 (4')^2 + 40320 (3')(5') + 40320 (2')(6')] - 5040 (1')^5(7')] * t^7 / 7!
P(8,t) = (1')^(-15) * [ -17297280 (2')^7 + 51891840 (1')(2')^5(3') - (1')^2 [39916800 (2')^3(3')^2 + 19958400 (2')^4(4')] + (1')^3 [6652800 (2')(3')^3 + 19958400 (2')^2(3')(4') + 6652800 (2')^3(5')] - (1')^4 [1814400 (3')^2(4') + 1814400 (2')(4')^2 + 3628800 (2')(3')(5') + 1814400 (2')^2(6')] + (1')^5 [362880 (4')(5') + 362880 (3')(6') + 362880 (2')(7')] - 40320 (1')^6(8')] * t^8 / 8!
...
See A134685 for more information.
A111785 is obtained from A133437 by dividing through the bracketed terms of the P(n,t) by n! and unsigned A111785 is a refinement of A033282 and A126216. - Tom Copeland, Sep 28 2008
For relation to the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see the Loday and McCammond links. E.g., P(5,t) = (1')^(-9) * [ 14 (2')^4 - 21 (1') (2')^2 (3') + 6 (1')^2 (2') (4')+ 3 (1')^2 (3')^2 - 1 (1')^3 (5') ] * t^5 is related to the 3-D associahedron with 14 vertices (0-D faces), 21 edges (1-D faces), 6 pentagons (2-D faces), 3 rectangles (2-D faces), 1 3-D polytope (3-D faces). Summing over faces of the same dimension gives A033282 or A126216. - Tom Copeland, Sep 29 2008
A relation between this Lagrange inversion for an o.g.f. and partition polynomials formed from the "refined Lah numbers" A130561 is presented in the link "Lagrange a la Lah" along with umbral binary tree representations.
With f(x,t) = x + t*Sum_{n>=2} u_n*x^n, the compositional inverse in x is related to the velocity profile of particles governed by the inviscid Burgers's, or Hopf, eqn. See A001764 and A086810. - Tom Copeland, Feb 15 2014
Newton was aware of this power series expansion for series reversion. See the Ferraro reference p. 75 eqn. 52. - Tom Copeland, Jan 22 2017
The coefficients of the partition polynomials divided by the associated factorial enumerate the faces of the convex, bounded polytopes called the associahedra, and the absolute value of the sum of the renormalized coefficients gives the Euler characteristic of unity for each polytope; i.e., the absolute value of the sum of each row of the array is either n! (unnormalized) or unity (normalized). In addition, the signs of the faces alternate with dimension, and the coefficients of faces with the same dimension for each polytope have the same sign. - Tom Copeland, Nov 13 2019
With u_1 = 1 and the other u_n replaced by suitably signed partition polynomials of A263633, the partition polynomials enumerating the refined noncrossing partitions of A134264 with a shift in indices are obtained (cf. In the Realm of Shadows). - Tom Copeland, Nov 16 2019
Relations between associahedra and oriented n-simplices are presented by Halvorson and by Street. - Tom Copeland, Dec 08 2019
Let f(x;t,n) = x - t * x^(n+1), giving u_1 = (1') = 1 and u_(n+1) = (n+1) = -t. Then inverting in x with t a parameter gives finv(x;t,n) = Sum_{j>=0} {binomial((n+1)*j,j) / (n*j + 1)} * t^j * x^(n*j + 1), which gives the Catalan numbers for n=1, and the Fuss-Catalan sequences for n>1 (see A001764, n=2). Comparing this with the same result in A134264 gives relations between the faces of associahedra and noncrossing partitions (and other combinatorial constructs related to these inversion formulas and those listed in A145271). - Tom Copeland, Jan 27 2020
From Tom Copeland, Mar 24 2020: (Start)
There is a mapping between the faces of K_n, the associahedron of dimension (n-1), and polygon dissections. The dissecting noncrossing diagonals (i.e., nonintersecting in the interior) form subpolygons. Assign the indeterminate x_k to a subpolygon where k = (number of vertices of the subpolygon) - 1. Multiply the x_k together to form the monomials for the inversion formula.
For the 3-dimensional associahedron K_4, the fundamental polygon is the hexagon, which can be dissected into pentagons, associated to x_4; tetragons, to x_3; and triangles, to x_2; for example, there are six distinguished partitions of the hexagon into one triangle and one pentagon, sharing two vertices, associated to the monomial 6 * x_2 * x_4 since the unshared vertex of the triangle can be moved consecutively from one vertex of the hexagon to the next. This term corresponds to 720 (1')^2 (2') (4') / 5! in P(5,t) above, denumerating the six pentagonal facets of K_4. (End)

References

  • G. Ferraro, The Rise and Development of the Theory of Series up to the Early 1820s, Springer Science and Business Media, 2007.
  • H. Halvorson (editor), Deep Beauty: Understanding the Quantum World Through Innovation, Cambridge Univ. Press, 2011.
  • H. Turnbull (editor), The Correspondence of Isaac Newton Vol. II 1676-1687, Cambridge Univ. Press, 1960, p. 147.

Links

Crossrefs

Cf. A145271, (A086810, A181289) = (reduced array, associated g(x)).
Cf. A001764, A007318, A033282, A086810, A111785, A126216, A129185.
Cf. A130561, A132159, A133314, A134264, A134685, A139605, A263633.

Programs

  • Mathematica
    rows[nn_] := {{1}}~Join~With[{s = InverseSeries[t (1 + Sum[u[k] t^k, {k, nn}] + O[t]^(nn+1))]}, Table[(n+1)! Coefficient[s, t^(n+1) Product[u[w], {w, p}]], {n, nn}, {p, Reverse[Sort[Sort /@ IntegerPartitions[n]]]}]];
rows[7] // Flatten (* Andrey Zabolotskiy, Mar 07 2024 *)

Formula

The bracketed partitions of P(n,t) are of the form (u_1)^e(1) (u_2)^e(2) ... (u_n)^e(n) with coefficients given by (-1)^(n-1+e(1)) * [2*(n-1)-e(1)]! / [ (e(2))! * (e(3))! * ... * (e(n))! ].
From Tom Copeland, Sep 06 2011: (Start)
Let h(t) = 1/(df(t)/dt)
= 1/Ev[u./(1-u.t)^2]
= 1/((u_1) + 2*(u_2)*t + 3*(u_3)*t^2 + 4*(u_4)*t^3 + ...),
where Ev denotes umbral evaluation.
Then for the partition polynomials of A133437,
n!*P(n,t) = ((t*h(y)*d/dy)^n) y evaluated at y=0,
and the compositional inverse of f(t) is
g(t) = exp(t*h(y)*d/dy) y evaluated at y=0.
Also, dg(t)/dt = h(g(t)). (End)
From Tom Copeland, Oct 20 2011: (Start)
With exp[x* PS(.,t)] = exp[t*g(x)] = exp[x*h(y)d/dy] exp(t*y) eval. at y=0, the raising/creation and lowering/annihilation operators defined by R PS(n,t)=PS(n+1,t) and L PS(n,t) = n*PS(n-1,t) are
R = t*h(d/dt) = t* 1/[(u_1) + 2*(u_2)*d/dt + 3*(u_3)*(d/dt)^2 + ...] and
L = f(d/dt) = (u_1)*d/dt + (u_2)*(d/dt)^2 + (u_3)*(d/dt)^3 + ....
Then P(n,t) = (t^n/n!) dPS(n,z)/dz eval. at z=0. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.) (End)
The bracketed partition polynomials of P(n,t) are also given by (d/dx)^(n-1) 1/[u_1 + u_2 * x + u_3 * x^2 + ... + u_n * x^(n-1)]^n evaluated at x=0. - Tom Copeland, Jul 07 2015
From Tom Copeland, Sep 20 2016: (Start)
Let PS(n,u1,u2,...,un) = P(n,t) / t^n, i.e., the square-bracketed part of the partition polynomials in the expansion for the inverse in the comment section, with u_k = uk.
Also let PS(n,u1=1,u2,...,un) = PB(n,b1,b2,...,bK,...) where each bK represents the partitions of PS, with u1 = 1, that have K components or blocks, e.g., PS(5,1,u2,...,u5) = PB(5,b1,b2,b3,b4) = b1 + b2 + b3 + b4 with b1 = -u5, b2 = 6 u2 u4 + 3 u3^2, b3 = -21 u2^2 u3, and b4 = 14 u2^4.
The relation between solutions of the inviscid Burgers' equation and compositional inverse pairs (cf. A086810) implies that, for n > 2, PB(n, 0 * b1, 1 * b2, ..., (K-1) * bK, ...) = [(n+1)/2] * Sum_{k = 2..n-1} PS(n-k+1,u_1=1,u_2,...,u_(n-k+1)) * PS(k,u_1=1,u_2,...,u_k).
For example, PB(5,0 * b1, 1 * b2, 2 * b3, 3 * b4) = 3 * 14 u2^4 - 2 * 21 u2^2 u3 + 1 * 6 u2 u4 + 1 * 3 u3^2 - 0 * u5 = 42 u2^4 - 42 u2^2 u3 + 6 u2 u4 + 3 u3^2 = 3 * [2 * PS(2,1,u2) * PS(4,1,u2,...,u4) + PS(3,1,u2,u3)^2] = 3 * [ 2 * (-u2) (-5 u2^3 + 5 u2 u3 - u4) + (2 u2^2 - u3)^2].
Also, PB(n,0*b1,1*b2,...,(K-1)*bK,...) = d/dt t^(n-2)*PS(n,u1=1/t,u2,...,un)|{t=1} = d/dt (1/t)*PS(n,u1=1,t*u2,...,t*un)|{t=1}.
(End)
From Tom Copeland, Sep 22 2016: (Start)
Equivalent matrix computation: Multiply the m-th diagonal (with m=1 the index of the main diagonal) of the lower triangular Pascal matrix A007318 by f_m = m!*u_m = (d/dx)^m f(x) evaluated at x=0 to obtain the matrix UP with UP(n,k) = binomial(n,k) f_{n+1-k}, or equivalently multiply the diagonals of A132159 by u_m. Then P(n,t) = (1, 0, 0, 0, ...) [UP^(-1) * S]^(n-1) FC * t^n/n!, where S is the shift matrix A129185, representing differentiation in the basis x^n//n!, and FC is the first column of UP^(-1), the inverse matrix of UP. These results follow from A145271 and A133314.
Also, P(n,t) = (1, 0, 0, 0, ...) [UP^(-1) * S]^n (0, 1, 0, ...)^T * t^n/n! in agreement with A139605. (End)
A recursion relation for computing each partition polynomial of this entry from the lower order polynomials and the coefficients of the refined Lah polynomials of A130561 is presented in the blog entry "Formal group laws and binomial Sheffer sequences." - Tom Copeland, Feb 06 2018
The derivative of the partition polynomials of A350499 with respect to a distinguished indeterminate give polynomials proportional to those of this entry. The connection of this derivative relation to the inviscid Burgers-Hopf evolution equation is given in a reference for that entry. - Tom Copeland, Feb 19 2022

Extensions

Missing coefficient in P(6,t) replaced by Tom Copeland, Nov 06 2008
P(7,t) and P(8,t) data added by Tom Copeland, Jan 14 2016
Title modified by Tom Copeland, Jan 13 2020
Terms ordered according to the reversed Abramowitz-Stegun ordering of partitions (with every k' replaced by (k-1)') by Andrey Zabolotskiy, Mar 07 2024

A133336 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 5, 1, 0, 14, 21, 9, 1, 0, 42, 84, 56, 14, 1, 0, 132, 330, 300, 120, 20, 1, 0, 429, 1287, 1485, 825, 225, 27, 1, 0, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 0, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 0, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1, 0
Offset: 0

Views

Author

Philippe Deléham, Oct 19 2007

Keywords

Comments

Mirror image of triangle A086810; another version of A126216.
Equals A131198*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007
Diagonal sums: A119370. - Philippe Deléham, Nov 09 2009

Examples

			Triangle begins:
    1;
    1,    0;
    2,    1,    0;
    5,    5,    1,   0;
   14,   21,    9,   1,   0;
   42,   84,   56,  14,   1,  0;
  132,  330,  300, 120,  20,  1, 0;
  429, 1287, 1485, 825, 225, 27, 1, 0;

Links

Crossrefs

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.

Programs

  • Magma
    [[Binomial(n-1,k)*Binomial(2*n-k,n)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 05 2018
  • Mathematica
    Table[Binomial[n-1,k]*Binomial[2*n-k,n]/(n+1), {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 05 2018 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(binomial(n-1,k)*binomial(2*n-k,n)/(n+1), ", "))) \\ G. C. Greubel, Feb 05 2018

Formula

Sum_{k=0..n} T(n,k)*x^k = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - Philippe Deléham, Nov 02 2009
Showing 1-10 of 17 results. Next

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