A001498 -id:A001498 - cp's OEIS frontend

A001147 Double factorial of odd numbers: a(n) = (2n-1)!! = 135...(2n-1).

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

Offset: 0

N. J. A. Sloane

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

			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 + ...

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Y. S. Song, On the combinatorics of rooted binary phylogenetic trees, Annals of Combinatorics, 7, 2003, 365-379. See Lemma 2.1. - _N. J. A. Sloane_, Aug 22 2014
D. Speyer and B. Sturmfels, The tropical Grassmannian, arXiv:0304218 [math.AG], 2003.
Neriman Tokcan, Jonathan Gryak, Kayvan Najarian, and Harm Derksen, Algebraic Methods for Tensor Data, arXiv:2005.12988 [math.RT], 2020.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).
Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Double Factorial
Eric Weisstein's World of Mathematics, Erf
Eric Weisstein's World of Mathematics, Ladder Rung Graph
Eric Weisstein's World of Mathematics, Normal Distribution Function
Wikipedia, Pfaffian
Wikipedia, Hermite polynomials
Index to divisibility sequences
Index entries for related partition-counting sequences
Index entries for sequences related to factorial numbers
Index entries for sequences related to parenthesizing
Index entries for "core" sequences

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.

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

a001147 n = product [1, 3 .. 2 * n - 1]
a001147_list = 1 : zipWith (*) [1, 3 ..] a001147_list
-- Reinhard Zumkeller, Feb 15 2015, Dec 03 2011

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

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

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

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 *)

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 */

{a(n) = if( n<0, (-1)^n / a(-n), (2*n)! / n! / 2^n)}; /* Michael Somos, Sep 18 2014 */

x='x+O('x^33); Vec(serlaplace((1-2*x)^(-1/2))) \\ Joerg Arndt, Apr 24 2011

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

[rising_factorial(n+1,n)/2^n for n in (0..15)] # Peter Luschny, Jun 26 2012

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

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

A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, 211799312, 997313824, 4809701440, 23758664096, 119952692896, 618884638912, 3257843882624, 17492190577600, 95680443760576, 532985208200576, 3020676745975552

Offset: 0

N. J. A. Sloane and J. H. Conway

a(n) is also the number of n X n symmetric permutation matrices.
a(n) is also the number of matchings (Hosoya index) in the complete graph K(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
a(n) is also the number of independent vertex sets and vertex covers in the n-triangular graph. - Eric W. Weisstein, May 22 2017
Equivalently, this is the number of graphs on n labeled nodes with degrees at most 1. - Don Knuth, Mar 31 2008
a(n) is also the sum of the degrees of the irreducible representations of the symmetric group S_n. - Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
a(n) is the number of partitions of a set of n distinguishable elements into sets of size 1 and 2. - Karol A. Penson, Apr 22 2003
Number of tableaux on the edges of the star graph of order n, S_n (sometimes T_n). - Roberto E. Martinez II, Jan 09 2002
The Hankel transform of this sequence is A000178 (superfactorials). Sequence is also binomial transform of the sequence 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, ... (A001147 with interpolated zeros). - Philippe Deléham, Jun 10 2005
Row sums of the exponential Riordan array (e^(x^2/2),x). - Paul Barry, Jan 12 2006
a(n) is the number of nonnegative lattice paths of upsteps U = (1,1) and downsteps D = (1,-1) that start at the origin and end on the vertical line x = n in which each downstep (if any) is marked with an integer between 1 and the height of its initial vertex above the x-axis. For example, with the required integer immediately preceding each downstep, a(3) = 4 counts UUU, UU1D, UU2D, U1DU. - David Callan, Mar 07 2006
Equals row sums of triangle A152736 starting with offset 1. - Gary W. Adamson, Dec 12 2008
Proof of the recurrence relation a(n) = a(n-1) + (n-1)*a(n-2): number of involutions of [n] containing n as a fixed point is a(n-1); number of involutions of [n] containing n in some cycle (j, n), where 1 <= j <= n-1, is (n-1) times the number of involutions of [n] containing the cycle (n-1 n) = (n-1)*a(n-2). - Emeric Deutsch, Jun 08 2009
Number of ballot sequences (or lattice permutations) of length n. A ballot sequence B is a string such that, for all prefixes P of B, h(i) >= h(j) for i < j, where h(x) is the number of times x appears in P. For example, the ballot sequences of length 4 are 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1231, and 1234. The string 1221 does not appear in the list because in the 3-prefix 122 there are two 2's but only one 1. (Cf. p. 176 of Bruce E. Sagan: "The Symmetric Group"). - Joerg Arndt, Jun 28 2009
Number of standard Young tableaux of size n; the ballot sequences are obtained as a length-n vector v where v_k is the (number of the) row in which the number r occurs in the tableaux. - Joerg Arndt, Jul 29 2012
Number of factorial numbers of length n-1 with no adjacent nonzero digits. For example the 10 such numbers (in rising factorial radix) of length 3 are 000, 001, 002, 003, 010, 020, 100, 101, 102, and 103. - Joerg Arndt, Nov 11 2012
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013
a(n) is the number of permutations of [n] that avoid the consecutive patterns 123 and 132. Proof. Write a self-inverse permutation in standard cycle form: smallest entry in each cycle in first position, first entries decreasing. For example, (6,7)(3,4)(2)(1,5) is in standard cycle form. Then erase parentheses. This is a bijection to the permutations that avoid consecutive 123 and 132 patterns. - David Callan, Aug 27 2014
Getu (1991) says these numbers are also known as "telephone numbers". - N. J. A. Sloane, Nov 23 2015
a(n) is the number of elements x in the symmetric group S_n such that x^2 = e where e is the identity. - Jianing Song, Aug 22 2018 [Edited on Jul 24 2025]
a(n) is the number of congruence orbits of upper-triangular n X n matrices on skew-symmetric matrices, or the number of Borel orbits in largest sect of the type DIII symmetric space SO_{2n}(C)/GL_n(C). Involutions can also be thought of as fixed-point-free partial involutions. See [Bingham and Ugurlu] link. - Aram Bingham, Feb 08 2020
From Thomas Anton, Apr 20 2020: (Start)
Apparently a(n) = b*c where b is odd iff a(n+b) (when a(n) is defined) is divisible by b.
Apparently a(n) = 2^(f(n mod 4)+floor(n/4))*q where f:{0,1,2,3}->{0,1,2} is given by f(0),f(1)=0, f(2)=1 and f(3)=2 and q is odd. (End)
From Iosif Pinelis, Mar 12 2021: (Start)
a(n) is the n-th initial moment of the normal distribution with mean 1 and variance 1. This follows because the moment generating function of that distribution is the e.g.f. of the sequence of the a(n)'s.
The recurrence a(n) = a(n-1) + (n-1)*a(n-2) also follows, by writing E(Z+1)^n=EZ(Z+1)^(n-1)+E(Z+1)^(n-1), where Z is a standard normal random variable, and then taking the first of the latter two integrals by parts. (End)

			Sequence starts 1, 1, 2, 4, 10, ... because possibilities are {}, {A}, {AB, BA}, {ABC, ACB, BAC, CBA}, {ABCD, ABDC, ACBD, ADCB, BACD, BADC, CBAD, CDAB, DBCA, DCBA}. - _Henry Bottomley_, Jan 16 2001
G.f. = 1 + x + 2*x^2 + 4*x^4 + 10*x^5 + 26*x^6 + 76*x^7 + 232*x^8 + 764*x^9 + ...
From _Gus Wiseman_, Jan 08 2021: (Start)
The a(4) = 10 standard Young tableaux:
  1 2 3 4
.
  1 2   1 3   1 2 3   1 2 4   1 3 4
  3 4   2 4   4       3       2
.
  1 2   1 3   1 4
  3     2     2
  4     4     3
.
  1
  2
  3
  4
The a(0) = 1 through a(4) = 10 set partitions into singletons or pairs:
  {}  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{3,4}}
             {{1},{2}}  {{1,2},{3}}    {{1,3},{2,4}}
                        {{1,3},{2}}    {{1,4},{2,3}}
                        {{1},{2},{3}}  {{1},{2},{3,4}}
                                       {{1},{2,3},{4}}
                                       {{1,2},{3},{4}}
                                       {{1},{2,4},{3}}
                                       {{1,3},{2},{4}}
                                       {{1,4},{2},{3}}
                                       {{1},{2},{3},{4}}
(End)

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Index entries for "core" sequences
Index entries for sequences related to Young tableaux.
Index entries for related partition-counting sequences

Cf. A001147, A001470, A053529, A099174, A136281-A136283, A001680, A001681.
See also A005425 for another version of the switchboard problem.
Equals 2 * A001475(n-1) for n>1.
First column of array A099020.
A069943(n+1)/A069944(n+1) = a(n)/A000142(n) in lowest terms.
Cf. A152736, A128229. - Gary W. Adamson, Dec 12 2008
Diagonal of A182172. - Alois P. Heinz, May 30 2012
Cf. A001813, A006882, A135401, A297708. - Manfred Scheucher, Jan 07 2018
Row sums of: A047884, A049403, A096713 (absolute value), A100861, A104556 (absolute value), A111924, A117506 (M_4 numbers), A122848, A238123.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A000700, A000701, A321729, A321730, A321737.

a000085 n = a000085_list !! n
  a000085_list = 1 : 1 : zipWith (+)
    (zipWith (*) [1..] a000085_list) (tail a000085_list) -- Reinhard Zumkeller, May 16 2013

A000085 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else procname(n-1)+(n-1)*procname(n-2); fi; end;
with(combstruct):ZL3:=[S,{S=Set(Cycle(Z,card<3))}, labeled]:seq(count(ZL3,size=n),n=0..25); # Zerinvary Lajos, Sep 24 2007
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(2):seq(count(A, size=n), n=0..25); # Zerinvary Lajos, Jun 11 2008

<Roger L. Bagula, Oct 06 2006 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2013 *)
a[ n_] := Sum[(2 k - 1)!! Binomial[ n, 2 k], {k, 0, n/2}]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, HypergeometricU[ -n/2, 1/2, -1/2] / (-1/2)^(n/2)]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x + x^2 / 2], {x, 0, n}]]; (* Michael Somos, Jun 01 2013 *)
Table[(I/Sqrt[2])^n HermiteH[n, -I/Sqrt[2]], {n, 0, 100}] (* Emanuele Munarini, Mar 02 2016 *)
a[n_] := Sum[StirlingS1[n, k]*2^k*BellB[k, 1/2], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 18 2017, after Emanuele Munarini *)
RecurrenceTable[{a[n] == a[n-1] + (n-1)*a[n-2], a[0] == 1, a[1] == 1}, a, {n, 0, 20}] (* Joan Ludevid, Jun 17 2022 *)
sds[{}]:={{}};sds[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sds[Complement[set,s]]]/@Cases[Subsets[set,{1,2}],{i,_}]; Table[Length[sds[Range[n]]],{n,0,10}] (* Gus Wiseman, Jan 11 2021 *)

B(n,x):=sum(stirling2(n,k)*x^k,k,0,n);
  a(n):=sum(stirling1(n,k)*2^k*B(k,1/2),k,0,n);
  makelist(a(n),n,0,40); /* Emanuele Munarini, May 16 2014 */

makelist((%i/sqrt(2))^n*hermite(n,-%i/sqrt(2)),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp( x + x^2 / 2 + x * O(x^n)), n))}; /* Michael Somos, Nov 15 2002 */

N=66; x='x+O('x^N); egf=exp(x+x^2/2); Vec(serlaplace(egf)) \\ Joerg Arndt, Mar 07 2013

from math import factorial
def A000085(n): return sum(factorial(n)//(factorial(n-(k<<1))*factorial(k)*(1<>1)+1)) # Chai Wah Wu, Aug 31 2023

A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
[simplify(A000085(n)) for n in range(28)] # Peter Luschny, Aug 21 2014

def a85(n): return sum(factorial(n) / (factorial(n-2*k) * 2**k * factorial(k)) for k in range(1+n//2))
for n in range(100): print(n, a85(n)) # Manfred Scheucher, Jan 07 2018

D-finite with recurrence a(0) = a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n>1.
E.g.f.: exp(x+x^2/2).
a(n) = a(n-1) + A013989(n-2) = A013989(n)/(n+1) = 1+A001189(n).
a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k*k!).
a(m+n) = Sum_{k>=0} k!*binomial(m, k)*binomial(n, k)*a(m-k)*a(n-k). - Philippe Deléham, Mar 05 2004
For n>1, a(n) = 2*(A000900(n) + A000902(floor(n/2))). - Max Alekseyev, Oct 31 2015
The e.g.f. y(x) satisfies y^2 = y''y' - (y')^2.
a(n) ~ c*(n/e)^(n/2)exp(n^(1/2)) where c=2^(-1/2)exp(-1/4). [Chowla]
a(n) = HermiteH(n, 1/(sqrt(2)*i))/(-sqrt(2)*i)^n, where HermiteH are the Hermite polynomials. - Karol A. Penson, May 16 2002
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006
For asymptotics see the Robinson paper.
a(n) = Sum_{m=0..n} A099174(n,m). - Roger L. Bagula, Oct 06 2006
O.g.f.: A(x) = 1/(1-x-1*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
From Gary W. Adamson, Dec 29 2008: (Start)
a(n) = (n-1)*a(n-2) + a(n-1); e.g., a(7) = 232 = 6*26 + 76.
Starting with offset 1 = eigensequence of triangle A128229. (End)
a(n) = (1/sqrt(2*Pi))*Integral_{x=-oo..oo} exp(-x^2/2)*(x+1)^n. - Groux Roland, Mar 14 2011
Row sums of |A096713|. a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A047974 and A080599. - Peter Bala, Dec 07 2011
From Sergei N. Gladkovskii, Dec 03 2011 - Oct 28 2013: (Start)
Continued fractions:
E.g.f.: 1+x*(2+x)/(2*G(0)-x*(2+x)) where G(k)=1+x*(x+2)/(2+2*(k+1)/G(k+1)).
G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 - x/U(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - x*(k+1)/Q(k+1)).
G.f.: -1/(x*Q(0)) where Q(k) = 1 - 1/x - (k+1)/Q(k+1).
G.f.: T(0)/(1-x) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)^2/T(k+1)). (End)
a(n) ~ (1/sqrt(2)) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014
a(n) = Sum_{k=0..n} s(n,k)*(-1)^(n-k)*2^k*B(k,1/2), where the s(n,k) are (signless) Stirling numbers of the first kind, and the B(n,x) = Sum_{k=0..n} S(n,k)*x^k are the Stirling polynomials, where the S(n,k) are the Stirling numbers of the second kind. - Emanuele Munarini, May 16 2014
a(n) = hyper2F0([-n/2,(1-n)/2],[],2). - Peter Luschny, Aug 21 2014
0 = a(n)*(+a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(-a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Aug 22 2014
From Peter Bala, Oct 06 2021: (Start)
a(n+k) == a(n) (mod k) for all n >= 0 and all positive odd integers k.
Hence for each odd k, the sequence obtained by taking a(n) modulo k is a periodic sequence and the exact period divides k. For example, taking a(n) modulo 7 gives the purely periodic sequence [1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, ...] of period 7. For similar results see A047974 and A115329. (End)

A003215 Hex (or centered hexagonal) numbers: 3n(n+1)+1 (crystal ball sequence for hexagonal lattice).

Original entry on oeis.org

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487, 6769

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Crystal ball sequence for A_2 lattice. - Michael Somos, Jun 03 2012
Sixth spoke of hexagonal spiral (cf. A056105-A056109).
Number of ordered integer triples (a,b,c), -n <= a,b,c <= n, such that a+b+c=0. - Benoit Cloitre, Jun 14 2003
Also the number of partitions of 6n into at most 3 parts, A001399(6n). - R. K. Guy, Oct 20 2003
Also, a(n) is the number of partitions of 6(n+1) into exactly 3 distinct parts. - William J. Keith, Jul 01 2004
Number of dots in a centered hexagonal figure with n+1 dots on each side.
Values of second Bessel polynomial y_2(n) (see A001498).
First differences of cubes (A000578). - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
Final digits of Hex numbers (hex(n) mod 10) are periodic with palindromic period of length 5 {1, 7, 9, 7, 1}. Last two digits of Hex numbers (hex(n) mod 100) are periodic with palindromic period of length 100. - Alexander Adamchuk, Aug 11 2006
All divisors of a(n) are congruent to 1, modulo 6. Proof: If p is an odd prime different from 3 then 3n^2 + 3n + 1 = 0 (mod p) implies 9(2n + 1)^2 = -3 (mod p), whence p = 1 (mod 6). - Nick Hobson, Nov 13 2006
For n>=1, a(n) is the side of Outer Napoleon Triangle whose reference triangle is a right triangle with legs (3a(n))^(1/2) and 3n(a(n))^(1/2). - Tom Schicker (tschicke(AT)email.smith.edu), Apr 25 2007
Number of triples (a,b,c) where 0<=(a,b)<=n and c=n (at least once the term n). E.g., for n = 1: (0,0,1), (0,1,0), (1,0,0), (0,1,1), (1,0,1), (1,1,0), (1,1,1), so a(1)=7. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007
Equals the triangular numbers convolved with [1, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009
From Terry Stickels, Dec 07 2009: (Start)
Also the maximum number of viewable cubes from any one static point while viewing a cube stack of identical cubes of varying magnitude.
For example, viewing a 2 X 2 X 2 stack will yield 7 maximum viewable cubes.
If the stack is 3 X 3 X 3, the maximum number of viewable cubes from any one static position is 19, and so on.
The number of cubes in the stack must always be the same number for width, length, height (at true regular cubic stack) and the maximum number of visible cubes can always be found by taking any cubic number and subtracting the number of the cube that is one less.
Examples: 125 - 64 = 61, 64 - 27 = 37, 27 - 8 = 19. (End)
The sequence of digital roots of the a(n) is period 3: repeat [1,7,1]. - Ant King, Jun 17 2012
The average of the first n (n>0) centered hexagonal numbers is the n-th square. - Philippe Deléham, Feb 04 2013
A002024 is the following array A read along antidiagonals:
  1,  2,  3,  4,  5,  6, ...
  2,  3,  4,  5,  6,  7, ...
  3,  4,  5,  6,  7,  8, ...
  4,  5,  6,  7,  8,  9, ...
  5,  6,  7,  8,  9, 10, ...
  6,  7,  8,  9, 10, 11, ...
and a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013
a(n) is the sum of the terms in the n+1 X n+1 matrices minus those in n X n matrices in an array formed by considering A158405 an array (the beginning terms in each row are 1,3,5,7,9,11,...). - J. M. Bergot, Jul 05 2013
The formula also equals the product of the three distinct combinations of two consecutive numbers: n^2, (n+1)^2, and n*(n+1). - J. M. Bergot, Mar 28 2014
The sides of any triangle ABC are divided into 2n + 1 equal segments by 2n points: A_1, A_2, ..., A_2n in side a, and also on the sides b and c cyclically. If A'B'C' is the triangle delimited by AA_n, BB_n and CC_n cevians, we have (ABC)/(A'B'C') = a(n) (see Java applet link). - Ignacio Larrosa Cañestro, Jan 02 2015
a(n) is the maximal number of parts into which (n+1) triangles can intersect one another. - Ivan N. Ianakiev, Feb 18 2015
((2^m-1)n)^t mod a(n) = ((2^m-1)(n+1))^t mod a(n) = ((2^m-1)(2n+1))^t mod a(n), where m any positive integer, and t = 0(mod 6). - Alzhekeyev Ascar M, Oct 07 2016
((2^m-1)n)^t mod a(n) = ((2^m-1)(n+1))^t mod a(n) = a(n) - (((2^m-1)(2n+1))^t mod a(n)), where m any positive integer, and t = 3(mod 6). - Alzhekeyev Ascar M, Oct 07 2016
(3n+1)^(a(n)-1) mod a(n) = (3n+2)^(a(n)-1) mod a(n) = 1. If a(n) not prime, then always strong pseudoprime. - Alzhekeyev Ascar M, Oct 07 2016
Every positive integer is the sum of 8 hex numbers (zero included), at most 3 of which are greater than 1. - Mauro Fiorentini, Jan 01 2018
Area enclosed by the segment of Archimedean spiral between n*Pi/2 and (n+1)*Pi/2 in Pi^3/48 units. - Carmine Suriano, Apr 10 2018
This sequence contains all numbers k such that 12*k - 3 is a square. - Klaus Purath, Oct 19 2021
The continued fraction expansion of sqrt(3*a(n)) is [3n+1; {1, 1, 2n, 1, 1, 6n+2}]. For n = 0, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 12 2022

			G.f. = 1 + 7*x + 19*x^2 + 37*x^3 + 61*x^4 + 91*x^5 + 127*x^6 + 169*x^7 + 217*x^8 + ...
From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms:
.
.                                 o o o o
.                   o o o        o o o o o
.         o o      o o o o      o o o o o o
.   o    o o o    o o o o o    o o o o o o o
.         o o      o o o o      o o o o o o
.                   o o o        o o o o o
.                                 o o o o
.
.   1      7          19             37
.
(End)
From _Klaus Purath_, Dec 03 2021: (Start)
(1) a(19) is not a prime number, because besides a(19) = a(9) + P(29), a(19) = a(15) + P(20) = a(2) + P(33) is also true.
(2) a(25) is prime, because except for a(25) = a(12) + P(38) there is no other equation of this pattern. (End)

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.
M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
G. L. Alexanderson and John E. Wetzel, Dissections of a tetrahedron, J. Combinatorial Theory Ser. B 11 (1971), 58--66. MR0303412 (46 #2549). See p. 58.
B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31 (see p. 30).
B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31. [Annotated scanned copy]
Aran Bingham, Commutative n-ary Arithmetic, University of New Orleans Theses and Dissertations, Paper 1959, 2015.
Henry Bottomley, Illustration of initial terms.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 41.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Dominique Désérable, Versatile Topology for Two-Dimensional Cellular Automata, Advances in Cellular Automata, Emergence, Complexity and Computation (ECC Vol 52) Springer, Cham (2025), Ch. 6, pp. 151-186.
M. Gardner & N. J. A. Sloane, Correspondence, 1973-74
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Md. Towhidul Islam, Extending triangle
G. S. Kazandzidis, On a Conjecture of Moessner and a General Problem, Bull. Soc. Math. Grèce, Nouvelle Série - vol. 2, fasc. 1-2, pp. 23-30 (1961).
Ignacio Larrosa Cañestro, Hexágono y estrella determinados por tres pares de cevianas simétricas, (java applet).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Vladimir Pletser, Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems, Preprints.org, 2024. See p. 20.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Hex Number
Eric Weisstein's World of Mathematics, Nexus Number
Eric Weisstein's World of Mathematics, Outer Napoleon Triangle.
Index entries for sequences related to centered polygonal numbers
Index entries for crystal ball sequences
Index entries for sequences related to A2 = hexagonal = triangular lattice
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000166, A000217, A000290, A000578 (the cubes, or partial sums), A001263, A001498, A002061, A002378, A002407 (primes), A003514, A005408, A005449, A005891, A028896, A048766, A056105, A056106, A056107, A056108, A056109, A063496, A056220, A130298, A132111 (second diagonal), A158405, A215630, A239449, A243201.
Column k=3 of A080853, and column k=2 of A047969.
See also A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.
Cf. A287326(A000124(n), 1).
Cf. A008292.
Cf. A005448, A140091, A001844, A000384, A060544, A045943.
Cf. A154105.

a003215 n = 3 * n * (n + 1) + 1  -- Reinhard Zumkeller, Oct 22 2011

[3*n*(n+1)+1: n in [0..50]]; // G. C. Greubel, Nov 04 2017

A003215:=n->3*n*(n+1)+1; seq(A003215(n), n=0..100); # Wesley Ivan Hurt, Mar 28 2014

FoldList[#1 + #2 &, 1, 6 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 7, 19}, 47] (* Robert G. Wilson v, Jul 06 2013 *)

makelist(3*n*(n+1)+1, n, 0, 30); /* Martin Ettl, Nov 12 2012 */

PARI
```
{a(n) = 3*n*(n+1) + 1};
```

[3*n*(n+1)+1 for n in range(47)] # Michael S. Branicky, Jan 07 2021

a(n) = 3*n*(n+1) + 1, n >= 0 (see the name).
a(n) = (n+1)^3 - n^3 = a(-1-n).
G.f.: (1 + 4*x + x^2) / (1 - x)^3. - Simon Plouffe in his 1992 dissertation
a(n) = 6*A000217(n) + 1.
a(n) = a(n-1) + 6*n = 2a(n-1) - a(n-2) + 6 = 3*a(n-1) - 3*a(n-2) + a(n-3) = A056105(n) + 5n = A056106(n) + 4*n = A056107(n) + 3*n = A056108(n) + 2*n = A056108(n) + n.
n-th partial arithmetic mean is n^2. - Amarnath Murthy, May 27 2003
a(n) = 1 + Sum_{j=0..n} (6*j). E.g., a(2)=19 because 1+ 6*0 + 6*1 + 6*2 = 19. - Xavier Acloque, Oct 06 2003
The sum of the first n hexagonal numbers is n^3. That is, Sum_{n>=1} (3*n*(n-1) + 1) = n^3. - Edward Weed (eweed(AT)gdrs.com), Oct 23 2003
a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 2 1 0 / 3 3 1]. M^n * [1 1 1] = [1 2n+1 a(n)]. E.g., a(4) = 61, right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 9 61] = [1 2n+1 a(4)]. - Gary W. Adamson, Dec 22 2004
Row sums of triangle A130298. - Gary W. Adamson, Jun 07 2007
a(n) = 3*n^2 + 3*n + 1. Proof: 1) If n occurs once, it may be in 3 positions; for the two other ones, n terms are independently possible, then we have 3*n^2 different triples. 2) If the term n occurs twice, the third one may be placed in 3 positions and have n possible values, then we have 3*n more different triples. 3) The term n may occurs 3 times in one way only that gives the formula. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007
Binomial transform of [1, 6, 6, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 6, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = (n-1)*A000166(n) + (n-2)*A000166(n-1) = (n-1)floor(n!*e^(-1)+1) + (n-2)*floor((n-1)!*e^(-1)+1) (with offset 0). - Gary Detlefs, Dec 06 2009
a(n) = A028896(n) + 1. - Omar E. Pol, Oct 03 2011
a(n) = integral( (sin((n+1/2)x)/sin(x/2))^3, x=0..Pi)/Pi. - Yalcin Aktar, Dec 03 2011
Sum_{n>=0} 1/a(n) = Pi/sqrt(3)*tanh(Pi/(2*sqrt(3))) = 1.305284153013581... - Ant King, Jun 17 2012
a(n) = A000290(n) + A000217(2n+1). - Ivan N. Ianakiev, Sep 24 2013
a(n) = A002378(n+1) + A056220(n) = A005408(n) + 2*A005449(n) = 6*A000217(n) + 1. - Ivan N. Ianakiev, Sep 26 2013
a(n) = 6*A000124(n) - 5. - Ivan N. Ianakiev, Oct 13 2013
a(n) = A239426(n+1) / A239449(n+1) = A215630(2*n+1,n+1). - Reinhard Zumkeller, Mar 19 2014
a(n) = A243201(n) / A002061(n + 1). - Mathew Englander, Jun 03 2014
a(n) = A101321(6,n). - R. J. Mathar, Jul 28 2016
E.g.f.: (1 + 6*x + 3*x^2)*exp(x). - Ilya Gutkovskiy, Jul 28 2016
a(n) = (A001844(n) + A016754(n))/2. - Bruce J. Nicholson, Aug 06 2017
a(n) = A045943(2n+1). - Miquel Cerda, Jan 22 2018
a(n) = 3*Integral_{x=n..n+1} x^2 dx. - Carmine Suriano, Apr 10 2018
a(n) = A287326(A000124(n), 1). - Kolosov Petro, Oct 22 2018
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=0} a(n)/n! = 10*e.
Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 2/e. (End)
G.f.: polylog(-3, x)*(1-x)/x. See the Simon Plouffe formula above, and the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 08 2021
a(n) = T(n-1)^2 - 2*T(n)^2 + T(n+1)^2, n >= 1, T = triangular number A000217. - Klaus Purath, Oct 11 2021
a(n) = 1 + 2*Sum_{j=n..2n} j. - Klaus Purath, Oct 19 2021
a(n) = A069099(n+1) - A000217(n). - Klaus Purath, Nov 03 2021
From Leo Tavares, Dec 03 2021: (Start)
a(n) = A005448(n) + A140091(n);
a(n) = A001844(n) + A002378(n);
a(n) = A005891(n) + A000217(n);
a(n) = A000290(n) + A000384(n+1);
a(n) = A060544(n-1) + 3*A000217(n);
a(n) = A060544(n-1) + A045943(n).
a(2*n+1) = A154105(n).
(End)

Partially edited by Joerg Arndt, Mar 11 2010

A008279 Triangle T(n,k) = n!/(n-k)! (0 <= k <= n) read by rows, giving number of permutations of n things k at a time.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 24, 1, 5, 20, 60, 120, 120, 1, 6, 30, 120, 360, 720, 720, 1, 7, 42, 210, 840, 2520, 5040, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 1, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880

Offset: 0

N. J. A. Sloane

Also called permutation coefficients.
Also falling factorials triangle A068424 with column a(n,0)=1 and row a(0,1)=1 otherwise a(0,k)=0, added. - Wolfdieter Lang, Nov 07 2003
The higher-order exponential integrals E(x,m,n) are defined in A163931; for information about the asymptotic expansion of E(x,m=1,n) see A130534. The asymptotic expansions for n = 1, 2, 3, 4, ..., lead to the right hand columns of the triangle given above. - Johannes W. Meijer, Oct 16 2009
The number of injective functions from a set of size k to a set of size n. - Dennis P. Walsh, Feb 10 2011
The number of functions f from {1,2,...,k} to {1,2,...,n} that satisfy f(x) >= x for all x in {1,2,...,k}. - Dennis P. Walsh, Apr 20 2011
T(n,k) = A181511(n,k) for k=1..n-1. - Reinhard Zumkeller, Nov 18 2012
The e.g.f.s enumerating the faces of the permutohedra / permutahedra, Perm(s,t;x) = [e^(sx)-1]/[s-t(e^(sx)-1)], (cf. A090582 and A019538) and the stellahedra / stellohedra, St(s,t;x) = [s e^((s+t)x)]/[s-t(e^(sx)-1)], (cf. A248727) given in Toric Topology satisfy exp[u*d/dt] St(s,t;x) = St(s,u+t;x) = [e^(ux)/(1-u*Perm(s,t;x))]*St(s,t;x), where e^(ux)/(1-uy) is a bivariate e.g.f. for the row polynomials of this entry and A094587. Equivalently, d/dt St = (x+Perm)*St and d/dt Perm = Perm^2, or d/dt log(St) = x + Perm and d/dt log(Perm) = Perm. - Tom Copeland, Nov 14 2016
T(n, k)/n! are the coefficients of the n-th exponential Taylor polynomial, or truncated exponentials, which was proved to be irreducible by Schur. See Coleman link. - Michel Marcus, Feb 24 2020
Given a generic choice of k+2 residues, T(n, k) is the number of meromorphic differentials on the Riemann sphere having a zero of order n and these prescribed residues at its k+2 poles. - Quentin Gendron, Jan 16 2025

			Triangle begins:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  6,   6;
  1,  4, 12,  24,   24;
  1,  5, 20,  60,  120,   120;
  1,  6, 30, 120,  360,   720,    720;
  1,  7, 42, 210,  840,  2520,   5040,   5040;
  1,  8, 56, 336, 1680,  6720,  20160,  40320,   40320;
  1,  9, 72, 504, 3024, 15120,  60480, 181440,  362880,  362880;
  1, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800;
  ...
For example, T(4,2)=12 since there are 12 injective functions f:{1,2}->{1,2,3,4}. There are 4 choices for f(1) and then, since f is injective, 3 remaining choices for f(2), giving us 12 ways to construct an injective function. - _Dennis P. Walsh_, Feb 10 2011
For example, T(5,3)=60 since there are 60 functions f:{1,2,3}->{1,2,3,4,5} with f(x) >= x. There are 5 choices for f(1), 4 choices for f(2), and 3 choices for f(3), giving us 60 ways to construct such a function. - _Dennis P. Walsh_, Apr 30 2011

CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 176; 31st ed., p. 215, Section 3.3.11.1.
Maple V Reference Manual, p. 490, numbperm(n,k).

T. D. Noe, Rows n = 0..100 of triangle, flattened
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013-2014.
V. Buchstaber and T. Panov Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.
R. F. Coleman, On the Galois groups of the exponential Taylor polynomials, L’Enseignement Mathématique 33, 183-189, (1987).
Quentin Gendron and Guillaume Tahar, Isoresidual fibration and resonance arrangements, Lett. Math. Phys. 112, No. 2, Paper No. 33, 36 p. (2022).
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients for k rooks on the 1xn board, all heights 1.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963) 31-41.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Yuriy Shablya and Dmitry Kruchinin, Euler-Catalan's Number Triangle and its Application, Proceedings Book of Micropam (The Mediterranean International Conference of Pure & Applied Mathematics and Related Areas 2018), 212-215.
Dennis Walsh, Notes on no-less functions
Eric Weisstein's World of Mathematics, Falling Factorial
Index entries for "core" sequences

Row sums give A000522.
Cf. A001497, A001498, A136572.
T(n,0)=A000012, T(n,1)=A000027, T(n+1,2)=A002378, T(n,3)=A007531, T(n,4)=A052762, and T(n,n)=A000142.
Cf. A019538, A090582, A094587, A248727.

a008279 n k = a008279_tabl !! n !! k
a008279_row n = a008279_tabl !! n
a008279_tabl = iterate f [1] where
   f xs = zipWith (+) ([0] ++ zipWith (*) xs [1..]) (xs ++ [0])
-- Reinhard Zumkeller, Dec 15 2013, Nov 18 2012

/* As triangle */ [[Factorial(n)/Factorial(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 11 2015

with(combstruct): for n from 0 to 10 do seq(count(Permutation(n),size=m), m = 0 .. n) od; # Zerinvary Lajos, Dec 16 2007
seq(seq(n!/(n-k)!,k=0..n),n=0..10); # Dennis P. Walsh, Apr 20 2011
seq(print(seq(pochhammer(n-k+1,k),k=0..n)),n=0..6); # Peter Luschny, Mar 26 2015

Table[CoefficientList[Series[(1 + x)^m, {x, 0, 20}], x]* Table[n!, {n, 0, m}], {m, 0, 10}] // Grid (* Geoffrey Critzer, Mar 16 2010 *)
Table[ Pochhammer[n - k + 1, k], {n, 0, 9}, {k, 0, n}] // Flatten (* or *)
Table[ FactorialPower[n, k], {n, 0, 9}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 18 2013, updated Jan 28 2016 *)

{T(n, k) = if( k<0 || k>n, 0, n!/(n-k)!)}; /* Michael Somos, Nov 14 2002 */

{T(n, k) = my(A, p); if( k<0 || k>n, 0, if( n==0, 1, A = matrix(n, n, i, j, x + (i==j)); polcoeff( sum(i=1, n!, if( p = numtoperm(n, i), prod(j=1, n, A[j, p[j]]))), k)))}; /* Michael Somos, Mar 05 2004 */

from math import factorial, isqrt, comb
def A008279(n): return factorial(a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))//factorial(a-n+comb(a+1,2)) # Chai Wah Wu, Nov 13 2024

for n in range(8): [falling_factorial(n,k) for k in (0..n)] # Peter Luschny, Mar 26 2015

E.g.f.: Sum T(n,k) x^n/n! y^k = exp(x)/(1-x*y). - Vladeta Jovovic, Aug 19 2002
Equals A007318 * A136572. - Gary W. Adamson, Jan 07 2008
T(n, k) = n*T(n-1, k-1) = k*T(n-1, k-1)+T(n-1, k) = n*T(n-1, k)/(n-k) = (n-k+1)*T(n, k-1). - Henry Bottomley, Mar 29 2001
T(n, k) = n!/(n-k)! if n >= k >= 0, otherwise 0.
G.f. for k-th column k!*x^k/(1-x)^(k+1), k >= 0.
E.g.f. for n-th row (1+x)^n, n >= 0.
Sum T(n, k)x^k = permanent of n X n matrix a_ij = (x+1 if i=j, x otherwise). - Michael Somos, Mar 05 2004
Ramanujan psi_1(k, x) polynomials evaluated at n+1. - Ralf Stephan, Apr 16 2004
E.g.f.: Sum T(n,k) x^n/n! y^k/k! = e^{x+xy}. - Franklin T. Adams-Watters, Feb 07 2006
The triangle is the binomial transform of an infinite matrix with (1, 1, 2, 6, 24, ...) in the main diagonal and the rest zeros. - Gary W. Adamson, Nov 20 2006
G.f.: 1/(1-x-xy/(1-xy/(1-x-2xy/(1-2xy/(1-x-3xy/(1-3xy/(1-x-4xy/(1-4xy/(1-... (continued fraction). - Paul Barry, Feb 11 2009
T(n,k) = Sum_{j=0..k} binomial(k,j)*T(x,j)*T(y,k-j) for x+y = n. - Dennis P. Walsh, Feb 10 2011
From Dennis P. Walsh, Apr 20 2011: (Start)
E.g.f (with k fixed): x^k*exp(x).
G.f. (with k fixed): k!*x^k/(1-x)^(k+1). (End)
For n >= 2 and m >= 2, Sum_{k=0..m-2} S2(n, k+2)*T(m-2, k) = Sum_{p=0..n-2} m^p. S2(n,k) are the Stirling numbers of the second kind A008277. - Tony Foster III, Jul 23 2019

A001515 Bessel polynomial y_n(x) evaluated at x=1.

Original entry on oeis.org

1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106, 36496226977, 841146804577, 21065166341402, 569600638022431, 16539483668991901, 513293594376771362, 16955228098102446847, 593946277027962411007, 21992967478132711654106, 858319677924203716921141

Offset: 0

N. J. A. Sloane

For some applications it is better to start this sequence with an extra 1 at the beginning: 1, 1, 2, 37, 266, 2431, 27007, 353522, 5329837, ... (again with offset 0). This sequence now has its own entry - see A144301.
Number of partitions of {1,...,k}, n <= k <= 2n, into n blocks with no more than 2 elements per block. Restated, number of ways to use the elements of {1,...,k}, n <= k <= 2n, once each to form a collection of n sets, each having 1 or 2 elements. - Bob Proctor, Apr 18 2005, Jun 26 2006. E.g., for n=2 we get: (k=2): {1,2}; (k=3): {1,23}, {2,13}, {3,12}; (k=4): {12,34}, {13,24}, {14,23}, for a total of a(2) = 7 partitions.
Equivalently, number of sequences of n unlabeled items such that each item occurs just once or twice (cf. A105749). - David Applegate, Dec 08 2008
Numerator of (n+1)-th convergent to 1+tanh(1). - Benoit Cloitre, Dec 20 2002
The following Maple lines show how this sequence and A144505, A144498, A001514, A144513, A144506, A144514, A144507, A144301 are related.
f0:=proc(n) local k; add((n+k)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f0(n),n=0..10)];
# that is this sequence
f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),n=0..10)];
# that is A144498
f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..10)];
# that is A144513; divided by 2 gives A001514
f3:=proc(n) local k; add((n+k+3)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f3(n),n=0..10)];
# that is A144514; divided by 6 gives A144506
f4:=proc(n) local k; add((n+k+4)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f4(n),n=0..10)];
# that divided by 24 gives A144507
a(n) is also the numerator of the continued fraction sequence beginning with 2 followed by 3 and the remaining odd numbers: [2,3,5,7,9,11,13,...]. - Gil Broussard, Oct 07 2009
Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most once. - N. J. A. Sloane, Jan 25 2017

			The first few Bessel polynomials are (cf. A001497, A001498):
  y_0 = 1
  y_1 = 1 +   x
  y_2 = 1 + 3*x +  3*x^2
  y_3 = 1 + 6*x + 15*x^2 + 15*x^3, etc.
G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 266*x^4 + 2431*x^5 + 27007*x^6 + 353522*x^7 + ...

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..404 (first 101 terms from T. D. Noe)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix I to "Analysis of the gift exchange problem", giving Type D recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix II to "Analysis of the gift exchange problem", giving Type C recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials, arXiv:quant-ph/0501155, 2005.
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 11.
O. Frink and H. L. Krall, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65,100-115, 1945. [From _Roger L. Bagula_, Feb 15 2009]
E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
Wojciech Mlotkowski and Anna Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
Index entries for sequences related to Bessel functions or polynomials
Index entries for related partition-counting sequences

Cf. A000108, A000806, A001514, A105749, A143990, A144505, A144506, A144507, A144513, A144514.
See A144301 for other formulas and comments.
Row sums of Bessel triangle A001497 as well as of A001498.
Partial sums: A105748.
First differences: A144498.
Replace "sets" with "lists" in comment: A001517.
The gift scenarios sequences when a gift can be stolen at most s times, for s = 1..9, are this sequence, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.

a001515 = sum . a001497_row -- Reinhard Zumkeller, Nov 24 2014

[(&+[Binomial(n+j, 2*j)*Catalan(j)*Factorial(j+1)/2^j: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Sep 26 2023

A001515 := proc(n) option remember; if n=0 then 1 elif n=1 then 2 else (2*n-1)*A001515(n-1)+A001515(n-2); fi; end;
A001515:=proc(n) local k; add( (n+k)!/((n-k)!*k!*2^k),k=0..n); end;
A001515:= n-> hypergeom( [n+1,-n],[],-1/2);
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==(2n-1)a[n-1]+a[n-2]},a[n], {n,25}] (* Harvey P. Dale, Jun 18 2011 *)
Table[Sum[BellY[n+1, k, (2 Range[n+1] - 3)!!], {k, n+1}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

{a(n) = if( n<0, n = -1 - n); sum( k=0, n, (2*n - k)! / (k! * (n-k)!) * 2^(k-n))} /* Michael Somos, Apr 08 2012 */

[sum(binomial(n+j,2*j)*binomial(2*j,j)*factorial(j)//2^j for j in range(n+1)) for n in range(31)] # G. C. Greubel, Sep 26 2023

The following formulas can all be found in (or are easily derived from formulas in) Grosswald's book.
D-finite with recurrence: a(0) = 1, a(1) = 2; thereafter a(n) = (2*n-1)*a(n-1) + a(n-2).
E.g.f.: exp(1-sqrt(1-2*x))/sqrt(1-2*x).
a(n) = Sum_{ k = 0..n } binomial(n+k,2*k)*(2*k)!/(k!*2^k).
Equivalently, a(n) = Sum_{ k = 0..n } (n+k)!/((n-k)!*k!*2^k) = Sum_{ k = n..2n } k!/((2n-k)!*(k-n)!*2^(k-n)).
a(n) = Hypergeometric2F0( [n+1, -n] ; - ; -1/2).
a(n) = A105749(n)/n!.
a(n) ~ exp(1)*(2n)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]
a(n) = A144301(n+1).
G.f.: 1/(1-x-x/(1-x-2*x/(1-x-3*x/(1-x-4*x/(1-x-5*x/(1-.... (continued fraction). - Paul Barry, Feb 08 2009
From Michael Somos, Apr 08 2012: (Start)
a(-1 - n) = a(n).
(a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. (End)
G.f.: 1/G(0) where G(k) = 1 - x - x*(2*k+1)/(1 - x - 2*x*(k+1)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2012
E.g.f.: E(0)/(2*sqrt(1-2*x)), where E(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)*(1+sqrt(1-2*x))/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013
G.f.: T(0)/(1-x), where T(k) = 1 - (k+1)*x/((k+1)*x - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2013
a(n) = (2*BesselI(1/2, 1)+BesselI(3/2, 1))*BesselK(n+1/2, 1). - Jean-François Alcover, Feb 03 2014
a(n) = exp(1)*sqrt(2/Pi)*BesselK(1/2+n,1). - Gerry Martens, Jul 22 2015
From Peter Bala, Apr 14 2017: (Start)
a(n) = (1/n!)*Integral_{x = 0..inf} exp(-x)*x^n*(1 + x/2)^n dx.
E.g.f.: d/dx( exp(x*c(x/2)) ) = 1 + 2*x + 7*x^2/2! + 37*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
From G. C. Greubel, Aug 16 2017: (Start)
a(n) = (1/2)_{n} * 2^n * hypergeometric1f1(-n; -2*n; 2).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 2*t/(1-t)^2). (End)

Extensively edited by N. J. A. Sloane, Dec 07 2008

A050534 Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n(n+1)(n-1)*(n-2)/8.

Original entry on oeis.org

0, 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, 2145, 3003, 4095, 5460, 7140, 9180, 11628, 14535, 17955, 21945, 26565, 31878, 37950, 44850, 52650, 61425, 71253, 82215, 94395, 107880, 122760, 139128, 157080, 176715, 198135, 221445, 246753, 274170, 303810, 335790

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

"There are n straight lines in a plane, no two of which are parallel and no three of which are concurrent. Their points of intersection being joined, show that the number of new lines drawn is (1/8)n(n-1)(n-2)(n-3)." (Schmall, 1915).
Several different versions of this sequence are possible, beginning with either one, two or three 0's.
If Y is a 3-subset of an n-set X then, for n>=6, a(n-4) is the number of (n-6)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Number of distinct ways to select 2 pairs of objects from a set of n+1 objects, when order doesn't matter. For example, with n = 3 (4 objects), the 3 possibilities are (12)(34), (13)(24), and (14)(23). - Brian Parsonnet, Jan 03 2012
Partial sums of A027480. - J. M. Bergot, Jul 09 2013
For the set {1,2,...,n}, the sum of the 2 smallest elements of all subsets with 3 elements is a(n) (see Bulut et al. link). - Serhat Bulut, Jan 20 2015
a(n) is also the number of subgroups of S_{n+1} (the symmetric group on n+1 elements) that are isomorphic to D_4 (the dihedral group of order 8). - Geoffrey Critzer, Sep 13 2015
a(n) is the coefficient of x1^(n-3)*x2^2 in exponential Bell polynomial B_{n+1}(x1,x2,...) (number of ways to select 2 pairs among n+1 objects, see above), hence its link with A000292 and A001296 (see formula). - Cyril Damamme, Feb 26 2018
Also the number of 4-cycles in the complete graph K_{n+1}. - Eric W. Weisstein, Mar 13 2018
Number of chiral pairs of colorings of the 4 edges or vertices of a square using n or fewer colors. Each member of a chiral pair is a reflection, but not a rotation, of the other. - Robert A. Russell, Oct 20 2020

			For a(3)=3, the chiral pairs of square colorings are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. - _Robert A. Russell_, Oct 20 2020

Arthur T. Benjamin and Jennifer Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 154.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, page 72.
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5, case k=2.

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, Jan 20 2015.
Alexander Burstein, Sergey Kitaev and Toufik Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A., Vol. 19, No. 2-3 (2008), pp. 27-38.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Frank Harary and Bennet Manvel, On the number of cycles in a graph, Matemat. casop. 21 (1971) 55-63, Theorem 1 for 4-cycles in complete graph.
Louis H. Kauffman, Non-Commutative Worlds-Classical Constraints, Relativity and the Bianchi Identity, arXiv preprint arXiv:1109.1085 [math-ph], 2011. (See Appendix)
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.2.
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, in Combinatorial Algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
C. N. Schmall, Problem 432, The American Mathematical Monthly, Vol. 22, No. 4 (1915), p. 130.
Eric Weisstein's World of Mathematics, Complete Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Tritriangular Number.
Chai Wah Wu, Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics, arXiv:1407.5663 [quant-ph], 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000332, A033487, A107394, A034827, A210569, Second column of triangle A001498.
Cf. similar sequences listed in A241765.
Cf. (square colorings) A006528 (oriented), A002817 (unoriented), A002411 (achiral),
Row 2 of A325006 (orthoplex facets, orthotope vertices) and A337409 (orthotope edges, orthoplex ridges).
Row 4 of A293496 (cycles of n colors using k or fewer colors).

List([0..40],n->3*Binomial(n+1,4)); # Muniru A Asiru, Mar 20 2018

[3*Binomial(n+1, 4): n in [0..40]]; // Vincenzo Librandi, Feb 14 2015

[seq(binomial(n+1,4)*3,n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Table[Binomial[Binomial[n, 2], 2], {n, 0, 50}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 3, 15}, 40] (* Harvey P. Dale, Dec 14 2011 *)
(* Start from Eric W. Weisstein, Mar 13 2018 *)
Binomial[Binomial[Range[0, 20], 2], 2]
Nest[Binomial[#, 2] &, Range[0, 20], 2]
Nest[PolygonalNumber[# - 1] &, Range[0, 20], 2]
CoefficientList[Series[3 x^3/(1 - x)^5, {x, 0, 20}], x]
(* End *)

a(n)=n*(n+1)*(n-1)*(n-2)/8 \\ Charles R Greathouse IV, Nov 20 2012

x='x+O('x^100); concat([0, 0, 0], Vec(3*x^3/(1-x)^5)) \\ Altug Alkan, Nov 01 2015

[(binomial(binomial(n,2),2)) for n in range(0, 39)] # Zerinvary Lajos, Nov 30 2009

a(n) = 3*binomial(n+1, 4) = 3*A000332(n+1).
From Vladeta Jovovic, May 03 2002: (Start)
Recurrence: a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 3*x^3 / (1-x)^5. (End)
a(n+1) = T(T(n)) - T(n); a(n+2) = T(T(n)+n) where T is A000217. - Jon Perry, Jun 11 2003
a(n+1) = T(n)^2 - T(T(n)) where T is A000217. - Jon Perry, Jul 23 2003
a(n) = T(T(n-1)-1) where T is A000217. - Jon E. Schoenfield, Dec 14 2014
a(n) = 3*C(n, 4) + 3*C(n, 3), for n>3.
From Alexander Adamchuk, Apr 11 2006: (Start)
a(n) = (1/2)*Sum_{k=1..n} k*(k-1)*(k-2).
a(n) = A033487(n-2)/2, n>1.
a(n) = C(n-1,2)*C(n+1,2)/2, n>2. (End)
a(n) = A052762(n+1)/8. - Zerinvary Lajos, Apr 26 2007
a(n) = (4x^4 - 4x^3 - x^2 + x)/2 where x = floor(n/2)*(-1)^n for n >= 0. - William A. Tedeschi, Aug 24 2010
E.g.f.: x^3*exp(x)*(4+x)/8. - Robert Israel, Nov 01 2015
a(n) = Sum_{k=1..n} Sum_{i=1..k} (n-i-1)*(n-k). - Wesley Ivan Hurt, Sep 12 2017
a(n) = A001296(n-1) - A000292(n-1). - Cyril Damamme, Feb 26 2018
Sum_{n>=3} 1/a(n) = 4/9. - Vaclav Kotesovec, May 01 2018
a(n) = A006528(n) - A002817(n) = (A006528(n) - A002411(n)) / 2 = A002817(n) - A002411(n). - Robert A. Russell, Oct 20 2020
Sum_{n>=3} (-1)^(n+1)/a(n) = 32*log(2)/3 - 64/9. - Amiram Eldar, Jan 09 2022
a(n) = Sum_{k=1..2} (-1)^(k+1)*binomial(n,2-k)*binomial(n,2+k). - Gerry Martens, Oct 09 2022

Additional comments from Antreas P. Hatzipolakis, May 03 2002

A100861 Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 6, 3, 1, 10, 15, 1, 15, 45, 15, 1, 21, 105, 105, 1, 28, 210, 420, 105, 1, 36, 378, 1260, 945, 1, 45, 630, 3150, 4725, 945, 1, 55, 990, 6930, 17325, 10395, 1, 66, 1485, 13860, 51975, 62370, 10395, 1, 78, 2145, 25740, 135135, 270270, 135135, 1, 91, 3003, 45045, 315315, 945945, 945945, 135135

Offset: 0

Emeric Deutsch, Jan 08 2005

Row n contains 1 + floor(n/2) terms. Row sums yield A000085. T(2n,n) = T(2n-1,n-1) = (2n-1)!! (A001147).
Inverse binomial transform is triangle with T(2n,n) = (2n-1)!!, 0 otherwise. - Paul Barry, May 21 2005
Equivalently, number of involutions of n with k pairs. - Franklin T. Adams-Watters, Jun 09 2006
From Gary W. Adamson, Dec 09 2009: (Start)
If considered as an infinite lower triangular matrix (cf. A144299),
lim_{n->} A100861^n = A118930: (1, 1, 2, 4, 13, 41, ...).
(End)
Sum_{k=0..floor(n/2)} T(n,k)m^(n-2k)s^(2k) is the n-th non-central moment of the normal probability distribution with mean m and standard deviation s. - Stanislav Sykora, Jun 19 2014
Row n is the list of coefficients of the independence polynomial of the n-triangular graph. - Eric W. Weisstein, Nov 11 2016
Restating the 2nd part of the Name, row n is the list of coefficients of the matching-generating polynomial of the complete graph K_n. - Eric W. Weisstein, Apr 03 2018

			T(4, 2) = 3 because in the graph with vertex set {A, B, C, D} and edge set {AB, BC, CD, AD, AC, BD} we have the following three 2-matchings: {AB, CD},{AC, BD} and {AD, BC}.
Triangle starts:
[0] 1;
[1] 1;
[2] 1,  1;
[3] 1,  3;
[4] 1,  6,   3;
[5] 1, 10,  15;
[6] 1, 15,  45,   15;
[7] 1, 21, 105,  105;
[8] 1, 28, 210,  420, 105;
[9] 1, 36, 378, 1260, 945.
.
From _Eric W. Weisstein_, Nov 11 2016: (Start)
As polynomials:
1,
1,
1 + x,
1 + 3*x,
1 + 6*x + 3*x^2,
1 + 10*x + 15*x^2,
1 + 15*x + 45*x^2 + 15*x^3. (End)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (1983 reprint), 10th edition, 1964, expression 22.3.11 in page 775.
C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

T. D. Noe, Rows n = 0..100, flattened
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
Gérard Le Caër, A new family of solvable Pearson-Dirichlet random walks, Journal of Statistical Physics 144:1 (2011), pp. 23-45.
Ji Young Choi and Jonathan D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016.
Mikael Fremling, On the modular covariance properties of composite fermions on the torus, arXiv:1810.10391 [cond-mat.str-el], 2018.
Gary R. W. Greaves, Jeven Syatriadi, and Charissa I. Utomo, Chromatic polynomials of signed graphs and dominating-vertex deletion formulae, arXiv:2407.00883 [math.CO], 2024. See p. 11.
A. Hernando, R. Hernando, A. Plastino and A. R. Plastino, The workings of the Maximum Entropy Principle in collective human behavior, arXiv preprint arXiv:1201.0905 [stat.AP], 2012.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
Aaron Pollack, Exceptional theta functions, arXiv:2211.05280 [math.NT], Nov 2022. See Lemma 7.5.1.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Triangular Graph
Wikipedia, Normal distribution, section 'Moments'
Jian Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Other versions of this same triangle are given in A144299, A001497, A001498, A111924.
Cf. A000085 (row sums).
Cf. A118930, A008619.

a100861 n k = a100861_tabf !! n !! k
a100861_row n = a100861_tabf !! n
a100861_tabf = zipWith take a008619_list a144299_tabl
-- Reinhard Zumkeller, Jan 02 2014

P[0]:=1: for n from 1 to 14 do P[n]:=sort(expand(P[n-1]+(n-1)*t*P[n-2])) od: for n from 0 to 14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields the sequence in triangular form
# Alternative:
A100861 := proc(n,k)
    n!/k!/(n-2*k)!/2^k ;
end proc:
seq(seq(A100861(n,k),k=0..n/2),n=0..10) ; # R. J. Mathar, Aug 19 2014

Table[Table[n!/(i! 2^i (n - 2 i)!), {i, 0, Floor[n/2]}], {n, 0, 10}] // Flatten  (* Geoffrey Critzer, Mar 27 2011 *)
CoefficientList[Table[2^(n/2) (-(1/x))^(-n/2) HypergeometricU[-n/2, 1/2, -1/(2 x)], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
CoefficientList[Table[(-I)^n Sqrt[x/2]^n HermiteH[n, I/Sqrt[2 x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)

T(n,k)=if(k<0 || 2*k>n, 0, n!/k!/(n-2*k)!/2^k) /* Michael Somos, Jun 04 2005 */

T(n, k) = n!/(k!(n-2k)!*2^k).
E.g.f.: exp(z+tz^2/2).
G.f.: g(t, z) satisfies the differential equation g = 1 + zg + tz^2*(d/dz)(zg).
Row generating polynomial = P[n] = [-i*sqrt(t/2)]^n*H(n, i/sqrt(2t)), where H(n, x) is a Hermite polynomial and i=sqrt(-1). Row generating polynomials P[n] satisfy P[0]=1, P[n] = P[n-1] + (n-1)tP[n-2].
T(n, k) = binomial(n, 2k)(2k-1)!!. - Paul Barry, May 21 2002 [Corrected by Roland Hildebrand, Mar 06 2009]
T(n,k) = (n-2k+1)*T(n-1,k-1) + T(n-1,k). - Franklin T. Adams-Watters, Jun 09 2006
E.g.f.: 1 + (x+y*x^2/2)/(E(0)-(x+y*x^2/2)), where E(k) = 1 + (x+y*x^2/2)/(1 + (k+1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 08 2013
T(n,k) = A144299(n,k), k=0..n/2. - Reinhard Zumkeller, Jan 02 2014

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.

Original entry on oeis.org

1, 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208, 30710464, 186753920, 1171979904, 7573069568, 50305536256, 342949298688, 2396286830080, 17138748412928, 125336396368896, 936222729254912, 7136574106003456, 55466948299223040, 439216305474605056, 3540846129311916032

Offset: 0

N. J. A. Sloane

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
Also the value of the n-th derivative of exp(x^2) evaluated at 1. - N. Calkin, Apr 22 2010
For n >= 1, a(n) is also the sum of the degrees of the irreducible representations of the group of n X n signed permutation matrices (described in sequence A066051). The similar sum for the "ordinary" symmetric group S_n is in sequence A000085. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 12 2002
It appears that this is also the number of permutations of 1, 2, ..., n+1 such that each term (after the first) is within 2 of some preceding term. Verified for n+1 <= 6. E.g., a(4) = 20 because of the 24 permutations of 1, 2, 3, 4, the only ones not permitted are 1, 4, 2, 3; 1, 4, 3, 2; 4, 1, 2, 3; and 4, 1, 3, 2. - Gerry Myerson, Aug 06 2003
Hankel transform is A108400. - Paul Barry, Feb 11 2008
From Emeric Deutsch, Jun 19 2010: (Start)
Number of symmetric involutions of [2n]. Example: a(2)=6 because we have 1234, 2143, 1324, 3412, 4231, and 4321. See the Egge reference, pp. 419-420.
Number of symmetric involutions of [2n+1]. Example: a(2)=6 because we have 12345, 14325, 21354, 45312, 52341, and 54321. See the Egge reference, pp. 419-420.
(End)
Binomial convolution of sequence A000085: a(n) = Sum_{k=0..n} binomial(n,k)*A000085(k)*A000085(n-k). - Emanuele Munarini, Mar 02 2016
The sequence can be obtained from the infinite product of 2 X 2 matrices [(1,N); (1,1)] by extracting the upper left terms, where N = (1, 3, 5, ...), the odd integers. - Gary W. Adamson, Jul 28 2016
Apparently a(n) is the number of standard domino tableaux of size 2n, where a domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. - Gus Wiseman, Feb 25 2018

			G.f. = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 312*x^5 + 1384*x^6 + 6512*x^7 + ...
The a(3) = 20 domino tableaux:
1 1 2 2 3 3
.
1 2 2 3 3
1
.
1 2 3 3   1 1 3 3   1 1 2 2
1 2       2 2       3 3
.
1 1 3 3   1 1 2 2
2         3
2         3
.
1 2 3   1 2 2   1 1 3
1 2 3   1 3 3   2 2 3
.
1 3 3   1 2 2
1       1
2       3
2       3
.
1 2   1 1   1 1
1 2   2 3   2 2
3 3   2 3   3 3
.
1 3   1 2   1 1
1 3   1 2   2 2
2     3     3
2     3     3
.
1 1
2
2
3
3
.
1
1
2
2
3
3 - _Gus Wiseman_, Feb 25 2018

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.1.4 Exer. 31.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
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).

T. D. Noe, Table of n, a(n) for n=0..200
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 10.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 10.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
R. A. Brualdi, Shi-Mie Ma, Enumeration of involutions by descents and symmetric matrices, Eur. J. Combin. 43 (2015) 220-228
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C.-O. Chow, Counting involutory, unimodal, and alternating signed permutations, Discr. Math. 306 (2006), 2222-2228.
R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv:1302.6150 [math.RT], 2013-2014.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Yen-chi R. Lin, Asymptotic Formula for Symmetric Involutions, arXiv:1310.0988 [math.CO], 2013.
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
D. P. Roberts and A. Venkatesh, Hurwitz monodromy and full number fields, 2014.
Index entries for sequences related to Hermite polynomials

Column k=2 of A294042 and A376826.
Row sums of A122832 and A185296.
Cf. A000085, A000712, A004003, A066051, A099390, A100861, A135401, A138178, A153452, A297388, A299699, A299926, A300056, A300060.

a000898 n = a000898_list !! n
a000898_list = 1 : 2 : (map (* 2) $
   zipWith (+) (tail a000898_list) (zipWith (*) [1..] a000898_list))
-- Reinhard Zumkeller, Oct 10 2011

# For Maple program see A000903.
seq(simplify((-I)^n*HermiteH(n, I)), n=0..25); # Peter Luschny, Oct 23 2015

a[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 17 2011, after Vladeta Jovovic *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2(a[n-1]+(n-1)a[n-2])},a,{n,30}] (* Harvey P. Dale, Aug 04 2012 *)
Table[Abs[HermiteH[n, I]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
a[ n_] := Sum[ 2^(n - 2 k) n! / (k! (n - 2 k)!), {k, 0, n/2}]; (* Michael Somos, Oct 23 2015 *)

makelist((%i)^n*hermite(n,-%i),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp(2*x + x^2 + x * O(x^n)), n))}; /* Michael Somos, Feb 08 2004 */

{a(n) = if( n<2, max(0, n+1), 2*a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Feb 08 2004 */

my(x='x+O('x^66)); Vec(serlaplace(exp(2*x+x^2))) \\ Joerg Arndt, Oct 04 2013

{a(n) = sum(k=0, n\2, 2^(n - 2*k) * n! / (k! * (n - 2*k)!))}; /* Michael Somos, Oct 23 2015 */

a(n) = Sum_{m=0..n} |A060821(n,m)| = H(n,-i)*i^n, with the Hermite polynomials H(n,x); i.e., these are row sums of the unsigned triangle A060821.
E.g.f.: exp(x*(x + 2)).
a(n) = 2 * A000902(n) for n >= 1.
a(n) = Sum_{k=0..n} binomial(n,2k)*binomial(2k,k)*k!*2^(n-2k). - N. Calkin, Apr 22 2010
Binomial transform of A047974. - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k). - Vladeta Jovovic, Oct 01 2003
From Paul Barry, Aug 29 2005: (Start)
a(n) = Sum_{k=0..floor(n/2)} A001498(n-k, k) * 2^(n-k).
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2) * 2^((n+k)/2) * (1+(-1)^(n-k))/2. (End)
For asymptotics, see the Robinson paper. [This is disputed by Yen-chi R. Lin. See below, Sep 30 2013.]
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * C(n,2*k) * (2*k)!/k!. - Paul Barry, Feb 11 2008
G.f.: 1/(1 - 2*x - 2*x^2/(1 - 2*x - 4*x^2/(1 - 2*x - 6*x^2/(1 - 2*x - 8*x^2/(1 - ... (continued fraction). - Paul Barry, Feb 25 2010
E.g.f.: exp(x^2 + 2*x) = Q(0); Q(k) = 1 + (x^2 + 2*x)/(2*k + 1 - (x^2 + 2*x)*(2*k + 1)/((x^2 + 2*x) + (2*k + 2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: 1/Q(0), where Q(k) = 1 + 2*x*k - x - x/(1 - 2*x*(k + 1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
a(n) = (2*n/e)^(n/2) * exp(sqrt(2*n)) / sqrt(2*e) * (1 + sqrt(2/n)/3 + O(n^(-1))). - Yen-chi R. Lin, Sep 30 2013
0 = a(n)*(2*a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-2*a(n+1) + a(n+2)) for all n >= 0. - Michael Somos, Oct 23 2015
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k)*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

More terms from Larry Reeves (larryr(AT)acm.org), Feb 21 2001
Initial condition a(0)=1 added to definition by Jon E. Schoenfield, Oct 01 2013
More terms from Joerg Arndt, Oct 04 2013

cp's OEIS Frontend

A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.

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A245066 Central terms of triangles A001497 and A001498.

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

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A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

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A003215 Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).

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A001147 Double factorial of odd numbers: a(n) = (2n-1)!! = 135...(2n-1).

A003215 Hex (or centered hexagonal) numbers: 3n(n+1)+1 (crystal ball sequence for hexagonal lattice).

A050534 Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n(n+1)(n-1)*(n-2)/8.