A000140 -id:A000140 - 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

A000124 Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379

Offset: 0

N. J. A. Sloane

These are Hogben's central polygonal numbers with the (two-dimensional) symbol
2
.P
1 n
The first line cuts the pancake into 2 pieces. For n > 1, the n-th line crosses every earlier line (avoids parallelism) and also avoids every previous line intersection, thus increasing the number of pieces by n. For 16 lines, for example, the number of pieces is 2 + 2 + 3 + 4 + 5 + ... + 16 = 137. These are the triangular numbers plus 1 (cf. A000217).
m = (n-1)(n-2)/2 + 1 is also the smallest number of edges such that all graphs with n nodes and m edges are connected. - Keith Briggs, May 14 2004
Also maximal number of grandchildren of a binary vector of length n+2. E.g., a binary vector of length 6 can produce at most 11 different vectors when 2 bits are deleted.
This is also the order dimension of the (strong) Bruhat order on the finite Coxeter group B_{n+1}. - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
Number of 132- and 321-avoiding permutations of {1,2,...,n+1}. - Emeric Deutsch, Mar 14 2002
For n >= 1 a(n) is the number of terms in the expansion of (x+y)*(x^2+y^2)*(x^3+y^3)*...*(x^n+y^n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003
Also the number of terms in (1)(x+1)(x^2+x+1)...(x^n+...+x+1); see A000140.
Narayana transform (analog of the binomial transform) of vector [1, 1, 0, 0, 0, ...] = A000124; using the infinite lower Narayana triangle of A001263 (as a matrix), N; then N * [1, 1, 0, 0, 0, ...] = A000124. - Gary W. Adamson, Apr 28 2005
Number of interval subsets of {1, 2, 3, ..., n} (cf. A002662). - Jose Luis Arregui (arregui(AT)unizar.es), Jun 27 2006
Define a number of straight lines in the plane to be in general arrangement when (1) no two lines are parallel, (2) there is no point common to three lines. Then these are the maximal numbers of regions defined by n straight lines in general arrangement in the plane. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
Note that a(n) = a(n-1) + A000027(n-1). This has the following geometrical interpretation: Suppose there are already n-1 lines in general arrangement, thus defining the maximal number of regions in the plane obtainable by n-1 lines and now one more line is added in general arrangement. Then it will cut each of the n-1 lines and acquire intersection points which are in general arrangement. (See the comments on A000027 for general arrangement with points.) These points on the new line define the maximal number of regions in 1-space definable by n-1 points, hence this is A000027(n-1), where for A000027 an offset of 0 is assumed, that is, A000027(n-1) = (n+1)-1 = n. Each of these regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n) = a(n-1) + A000027(n-1). Cf. the comments on A000125 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
When constructing a zonohedron, one zone at a time, out of (up to) 3-d non-intersecting parallelepipeds, the n-th element of this sequence is the number of edges in the n-th zone added with the n-th "layer" of parallelepipeds. (Verified up to 10-zone zonohedron, the enneacontahedron.) E.g., adding the 10th zone to the enneacontahedron requires 46 parallel edges (edges in the 10th zone) by looking directly at a 5-valence vertex and counting visible vertices. - Shel Kaphan, Feb 16 2006
Binomial transform of (1, 1, 1, 0, 0, 0, ...) and inverse binomial transform of A072863: (1, 3, 9, 26, 72, 192, ...). - Gary W. Adamson, Oct 15 2007
If Y is a 2-subset of an n-set X then, for n >= 3, a(n-3) is the number of (n-2)-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Equals row sums of triangle A144328. - Gary W. Adamson, Sep 18 2008
It appears that a(n) is the number of distinct values among the fractions F(i+1)/F(j+1) as j ranges from 1 to n and, for each fixed j, i ranges from 1 to j, where F(i) denotes the i-th Fibonacci number. - John W. Layman, Dec 02 2008
a(n) is the number of subsets of {1,2,...,n} that contain at most two elements. - Geoffrey Critzer, Mar 10 2009
For n >= 2, a(n) gives the number of sets of subsets A_1, A_2, ..., A_n of n = {1, 2, ..., n} such that Meet_{i = 1..n} A_i is empty and Sum_{j in [n]} (|Meet{i = 1..n, i != j} A_i|) is a maximum. - Srikanth K S, Oct 22 2009
The numbers along the left edge of Floyd's triangle. - Paul Muljadi, Jan 25 2010
Let A be the Hessenberg matrix of order n, defined by: A[1,j] = A[i,i]:=1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n-1) = (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 24 2010
Also the number of deck entries of Euler's ship. See the Meijer-Nepveu link. - Johannes W. Meijer, Jun 21 2010
(1 + x^2 + x^3 + x^4 + x^5 + ...)*(1 + 2x + 3x^2 + 4x^3 + 5x^4 + ...) = (1 + 2x + 4x^2 + 7x^3 + 11x^4 + ...). - Gary W. Adamson, Jul 27 2010
The number of length n binary words that have no 0-digits between any pair of consecutive 1-digits. - Jeffrey Liese, Dec 23 2010
Let b(0) = b(1) = 1; b(n) = max(b(n-1)+n-1, b(n-2)+n-2) then a(n) = b(n+1). - Yalcin Aktar, Jul 28 2011
Also number of triangular numbers so far, for n > 0: a(n) = a(n-1) + Sum(A010054(a(k)): 0 <= k < n), see also A097602, A131073. - Reinhard Zumkeller, Nov 15 2012
Also number of distinct sums of 1 through n where each of those can be + or -. E.g., {1+2,1-2,-1+2,-1-2} = {3,-1,1,-3} and a(2) = 4. - Toby Gottfried, Nov 17 2011
This sequence is complete because the sum of the first n terms is always greater than or equal to a(n+1)-1. Consequently, any nonnegative number can be written as a sum of distinct terms of this sequence. See A204009, A072638. - Frank M Jackson, Jan 09 2012
The sequence is the number of distinct sums of subsets of the nonnegative integers, and its first differences are the positive integers. See A208531 for similar results for the squares. - John W. Layman, Feb 28 2012
Apparently the number of Dyck paths of semilength n+1 in which the sum of the first and second ascents add to n+1. - David Scambler, Apr 22 2013
Without 1 and 2, a(n) equals the terminus of the n-th partial sum of sequence 1, 1, 2. Explanation: 1st partial sums of 1, 1, 2 are 1, 2, 4; 2nd partial sums are 1, 3, 7; 3rd partial sums are 1, 4, 11; 4th partial sums are 1, 5, 16, etc. - Bob Selcoe, Jul 04 2013
Equivalently, numbers of the form 2*m^2+m+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - Bruno Berselli, Apr 08 2014
For n >= 2: quasi-triangular numbers; the almost-triangular numbers being A000096(n), n >= 2. Note that 2 is simultaneously almost-triangular and quasi-triangular. - Daniel Forgues, Apr 21 2015
n points in general position determine "n choose 2" lines, so A055503(n) <= a(n(n-1)/2). If n > 3, the lines are not in general position and so A055503(n) < a(n(n-1)/2). - Jonathan Sondow, Dec 01 2015
The digital root is period 9 (1, 2, 4, 7, 2, 7, 4, 2, 1), also the digital roots of centered 10-gonal numbers (A062786), for n > 0, A133292. - Peter M. Chema, Sep 15 2016
Partial sums of A028310. - J. Conrad, Oct 31 2016
For n >= 0, a(n) is the number of weakly unimodal sequences of length n over the alphabet {1, 2}. - Armend Shabani, Mar 10 2017
From Eric M. Schmidt, Jul 17 2017: (Start)
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) != e(k). [Martinez and Savage, 2.4]
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) and e(i) < e(k). [Martinez and Savage, 2.4]
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(j) != e(k). [Martinez and Savage, 2.4]
(End)
Numbers m such that 8m - 7 is a square. - Bruce J. Nicholson, Jul 24 2017
From Klaus Purath, Jan 29 2020: (Start)
The odd prime factors != 7 occur in an interval of p successive terms either never or exactly twice, while 7 always occurs only once. If a prime factor p appears in a(n) and a(m) within such an interval, then n + m == -1 (mod p). When 7 divides a(n), then 2*n == -1 (mod 7). a(n) is never divisible by the prime numbers given in A003625.
While all prime factors p != 7 can occur to any power, a(n) is never divisible by 7^2. The prime factors are given in A045373. The prime terms of this sequence are given in A055469.
(End)
From Roger Ford, May 10 2021: (Start)
a(n-1) is the greatest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
      /\   The top arch has a length of 3.  /\   The top arch has a length of 3.
     /  \  Both bottom arches have a       //\\  The middle arch has a length of 2.
    //\/\\ length of 1.                   ///\\\ The bottom arch has a length of 1.
Example: for n = 4, a(4-1) = a(3) = 7     /\
                                         //\\
                                     /\ ///\\\   1 + 3 + 2 + 1 = 7.  (End)
a(n+1) is the a(n)-th smallest positive integer that has not yet appeared in the sequence. - Matthew Malone, Aug 26 2021
For n> 0, let the n-dimensional cube {0,1}^n be, provided with the Hamming distance, d. Given an element x in {0,1}^n, a(n) is the number of elements y in {0,1}^n such that d(x, y) <= 2. Example: n = 4. (0,0,0,0), (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0) are at distance <= 2 from (0,0,0,0), so a(4) = 11. - Yosu Yurramendi, Dec 10 2021
a(n) is the sum of the first three entries of row n of Pascal's triangle. - Daniel T. Martin, Apr 13 2022
a(n-1) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 3 with exactly one descent. For example, sigma is one of the patterns, {132, 213, 231, 312}. - Jessica A. Tomasko, Sep 14 2022
a(n+4) is the number of ways to tile an equilateral triangle of side length 2*n with smaller equilateral triangles of side length n and side length 1. For example, with n=2, there are 22 ways to tile an equilateral triangle of side length 4 with smaller ones of sides 2 and 1, including the one tiling with sixteen triangles of sides 1 and the one tiling with four triangles of sides 2. - Ahmed ElKhatib and Greg Dresden, Aug 19 2024
Define a "hatpin" to be the planar graph consisting of a distinguished point (called the "head") and a semi-infinite line from that point. The maximum number of regions than can be formed by drawing n hatpins is a(n-1). See link for the case n = 4. - N. J. A. Sloane, Jun 25 2025

			a(3) = 7 because the 132- and 321-avoiding permutations of {1, 2, 3, 4} are 1234, 2134, 3124, 2314, 4123, 3412, 2341.
G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 22*x^6 + 29*x^7 + ...

Robert B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 24.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 80.
Henry Ernest Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 98.
William Allen Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
Akiva M. Yaglom and Isaak M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #44 (First published: San Francisco: Holden-Day, Inc., 1964).

T. D. Noe, Table of n, a(n) for n = 0..1000
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Jean-Luc Baril and Céline Moreira Dos Santos, Pizza-cutter's problem and Hamiltonian path, Mathematics Magazine (2019) Vol. 88, No. 1, 1-9.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Jean-Luc Baril, Toufik Mansour, and Armen Petrossian, Equivalence classes of permutations modulo excedances, preprint, Journal of Combinatorics, Volume 5 (2014) Number 4.
Jean-Luc Baril and Armen Petrossian, Equivalence classes of permutations modulo descents and left-to-right maxima, preprint, Pure Mathematics and Applications, Volume 25, Issue 1 (Sep 2015).
Andrew M. Baxter and Lara K. Pudwell, Ascent sequences avoiding pairs of patterns, preprint, The Electronic Journal of Combinatorics, Volume 22, Issue 1 (2015) Paper #P1.58.
Christian Bean, Anders Claesson, and Henning Ulfarsson, Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3, arXiv preprint arXiv:1512.03226 [math.CO], 2017.
Henry Bottomley, Illustration of initial terms.
Alexander Burstein and Toufik Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001.
Alexander Burstein and Toufik Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14.
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 45, 56.
Peter M. Chema, Illustration of first 22 terms as corners of a double square spiral with digital root.
David Coles, Triangle Puzzle.
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
Tom Crawford, 22 Slices of Pizza with Six Cuts, Tom Rocks Maths video (2022)
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
Karl Dilcher and Kenneth B. Stolarsky, Nonlinear recurrences related to Chebyshev polynomials, The Ramanujan Journal, 2014, Online Oct. 2014, pp. 1-23. See Cor. 5.
Igor Dolinka, James East, and Robert D. Gray, Motzkin monoids and partial Brauer monoids, Journal of Algebra, volume 471, February 2017, pages 251-298. Also preprint arXiv:1512.02279 [math.GR], 2015. See Table 5.
Matthew England, Russell Bradford, and James H. Davenport, Cylindrical algebraic decomposition with equational constraints, Journal of Symbolic Computation, Vol. 100 (2020), pp. 38-71; arXiv preprint, arXiv:1903.08999 [cs.SC], 2019.
J. B. Gil and J. Tomasko, Restricted Grassmannian permutations, ECA 2:4 (2022) Article S4PP6.
Sahir Gill, Bounds for Region Containing All Zeros of a Complex Polynomial, International Journal of Mathematical Analysis (2018), Vol. 12, No. 7, 325-333.
Richard K. Guy, Letter to N. J. A. Sloane.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
M. F. Hasler, Interactive illustration of A000124. [Sep 06 2017: The user can choose the slices to make, but the program can suggest a set of n slices which should yield the maximum number of pieces. For n slices this obviously requires 2n endpoints, or 2n+1 if they are equally spaced, so if there are not enough "blobs", their number is accordingly increased. This is the distinction between "draw" (done when you change the slices or number of blobs by hand) and "suggest" (propose a new set of slices).]
Phillip Tomas Heikoop, Dimensions of Matrix Subalgebras, Bachelor's Thesis, Worcester Polytechnic Institute, Massachusetts, 2019.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Cheyne Homberger and Vincent Vatter, On the effective and automatic enumeration of polynomial permutation classes, Journal of Symbolic Computation, Vol. 76 (2016), pp. 84-96; arXiv preprint, arXiv:1308.4946 [math.CO], 2013-2015.
Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 22.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 386
Milan Janjic, Two Enumerative Functions.
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Myrto Kallipoliti, Robin Sulzgruber, and Eleni Tzanaki, Patterns in Shi tableaux and Dyck paths, arXiv:2006.06949 [math.CO], 2020.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Thomas Langley, Jeffrey Liese, and Jeffrey Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016-2017.
Derek Levin, Lara Pudwell, Manda Riehl and Andrew Sandberg, Pattern Avoidance on k-ary Heaps, Slides of Talk, 2014.
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
Jim Loy, Triangle Puzzle.
Toufik Mansour, Permutations avoiding a set of patterns from S_3 and a pattern from S_4, arXiv:math/9909019 [math.CO], 1999.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016-2018.
Johannes W. Meijer and Manuel Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187.
Markus Moll, On a family of random noble means substitutions, Dr. Math. Dissertation, Universität Bielefeld, 2013, arXiv:1312.5136 [math.DS], 2013.
Permutation Pattern Avoidance Library (PermPAL), Av(123,231)
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
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Lara Pudwell and Andrew Baxter, Ascent sequences avoiding pairs of patterns, 2014.
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Franck Ramaharo and Fanja Rakotondrajao, A state enumeration of the foil knot, arXiv:1712.04026 [math.CO], 2017.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
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Nathan Reading, Order Dimension, Strong Bruhat Order and Lattice Properties for Posets.
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Herman P. Robinson, Letter to N. J. A. Sloane, Aug 16 1971, with attachments.
Rodica Simion and Frank W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985; see Example 3.5.
N. J. A. Sloane, Four hatpins can divide the plane into a(3) = 7 regions.
N. J. A. Sloane, On single-deletion-correcting codes, 2002.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 1.
Andrew James Turner and Julian Francis Miller, Recurrent Cartesian Genetic Programming Applied to Famous Mathematical Sequences, 2014.
Eric Weisstein's World of Mathematics, Circle Division by Lines.
Eric Weisstein's World of Mathematics, Plane Division by Lines.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, Vol. 8 (2008), pp. 45-52.
Wikipedia, Floyd's triangle.
Index entries for "core" sequences.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096 (Maximal number of pieces that can be obtained by cutting an annulus with n cuts, for n >= 1).
Slicing a cake: A000125, a bagel: A003600.
Partial sums =(A033547)/2, (A014206)/2.
The first 20 terms are also found in A025732 and A025739.
Cf. A002061, A002522, A016028, A055503, A072863, A144328, A177862, A263883, A000127, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A051601, A228918.
Cf. also A055469 Quasi-triangular primes, A002620, A000217.
A row of the array in A386478.

List([0..60],n->n*(n+1)/2+1); # Muniru A Asiru, Apr 11 2018

a000124 = (+ 1) . a000217
-- Reinhard Zumkeller, Oct 04 2012, Nov 15 2011

[n: n in [0..1500] | IsSquare(8*n-7)]; // Vincenzo Librandi, Apr 16 2014

Maple
```
A000124 := n-> n*(n+1)/2+1;
```

FoldList[#1 + #2 &, 1, Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[Range[0, 60]] + 1 (* Harvey P. Dale, Mar 12 2013 *)
Select[Range[2000], IntegerQ[Sqrt[8 # - 7]] &] (* Vincenzo Librandi, Apr 16 2014 *)
Table[PolygonalNumber[n] + 1, {n, 0, 52}] (* Michael De Vlieger, Jun 30 2016, Version 10.4 *)
LinearRecurrence[{3, -3, 1}, {1, 2, 4}, 53] (* Jean-François Alcover, Sep 23 2017 *)

{a(n) = (n^2 + n) / 2 + 1}; /* Michael Somos, Sep 04 2006 */

def a(n): return n*(n+1)//2 + 1
print([a(n) for n in range(53)]) # Michael S. Branicky, Aug 26 2021

(1 to 52).scanLeft(1)( + ) // Alonso del Arte, Feb 24 2019

G.f.: (1 - x + x^2)/(1 - x)^3. - Simon Plouffe in his 1992 dissertation
a(n) = A108561(n+3, 2). - Reinhard Zumkeller, Jun 10 2005
G.f.: (1 - x^6)/((1 - x)^2*(1 - x^2)*(1 - x^3)). a(n) = a(-1 - n) for all n in Z. - Michael Somos, Sep 04 2006
Euler transform of length 6 sequence [ 2, 1, 1, 0, 0, -1]. - Michael Somos, Sep 04 2006
a(n+3) = 3*a(n+2) - 3*a(n+1) + a(n) and a(1) = 1, a(2) = 2, a(3) = 4. - Artur Jasinski, Oct 21 2008
a(n) = A000217(n) + 1.
a(n) = a(n-1) + n. E.g.f.:(1 + x + x^2/2)*exp(x). - Geoffrey Critzer, Mar 10 2009
a(n) = Sum_{k = 0..n + 1} binomial(n+1, 2(k - n)). - Paul Barry, Aug 29 2004
a(n) = binomial(n+2, 1) - 2*binomial(n+1, 1) + binomial(n+2, 2). - Zerinvary Lajos, May 12 2006
From Thomas Wieder, Feb 25 2009: (Start)
a(n) = Sum_{l_1 = 0..n + 1} Sum_{l_2 = 0..n}...Sum_{l_i = 0..n - i}...Sum_{l_n = 0..1} delta(l_1, l_2, ..., l_i, ..., l_n) where delta(l_1, l_2, ..., l_i, ..., l_n) = 0 if any l_i != l_(i+1) and l_(i+1) != 0 and delta(l_1, l_2, ..., l_i, ..., l_n) = 1 otherwise. (End)
a(n) = A034856(n+1) - A005843(n) = A000217(n) + A005408(n) - A005843(n). - Jaroslav Krizek, Sep 05 2009
a(n) = 2*a(n-1) - a(n-2) + 1. - Eric Werley, Jun 27 2011
E.g.f.: exp(x)*(1+x+(x^2)/2) = Q(0); Q(k) = 1+x/(1-x/(2+x-4/(2+x*(k+1)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
a(n) = A014132(n, 1) for n > 0. - Reinhard Zumkeller, Dec 12 2012
a(n) = 1 + floor(n/2) + ceiling(n^2/2) = 1 + A004526(n) + A000982(n). - Wesley Ivan Hurt, Jun 14 2013
a(n) = A228074(n+1, n). - Reinhard Zumkeller, Aug 15 2013
For n > 0: A228446(a(n)) = 3. - Reinhard Zumkeller, Mar 12 2014
a(n) >= A263883(n) and a(n(n-1)/2) >= A055503(n). - Jonathan Sondow, Dec 01 2015
From Ilya Gutkovskiy, Jun 29 2016: (Start)
Dirichlet g.f.: (zeta(s-2) + zeta(s-1) + 2*zeta(s))/2.
Sum_{n >= 0} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) = A226985. (End)
a(n) = (n+1)^2 - A000096(n). - Anton Zakharov, Jun 29 2016
a(n) = A101321(1, n). - R. J. Mathar, Jul 28 2016
a(n) = 2*a(n-1) - binomial(n-1, 2) and a(0) = 1. - Armend Shabani, Mar 10 2017
a(n) = A002620(n+2) + A002620(n-1). - Anton Zakharov, May 11 2017
From Klaus Purath, Jan 29 2020: (Start)
a(n) = (Sum_{i=n-2..n+2} A000217(i))/5.
a(n) = (Sum_{i=n-2..n+2} A002378(i))/10.
a(n) = (Sum_{i=n..n+2} A002061(i)+1)/6.
a(n) = (Sum_{i=n-1..n+2} A000290(i)+2)/8.
a(n) = A060533(n-1) + 10, n > 5.
a(n) = (A002378(n) + 2)/2.
a(n) = A152948(n+2) - 1.
a(n) = A152950(n+1) - 2.
a(n) = (A002061(n) + A002061(n+2))/4.
(End)
Sum_{n>=0} (-1)^n/a(n) = A228918. - Amiram Eldar, Nov 20 2020
From Amiram Eldar, Feb 17 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(15)*Pi/2)*sech(sqrt(7)*Pi/2).
Product_{n>=1} (1 - 1/a(n)) = 2*Pi*sech(sqrt(7)*Pi/2). (End)
a((n^2-3n+6)/2) + a((n^2-n+4)/2) = a(n^2-2n+6)/2. - Charlie Marion, Feb 14 2023

A008302 Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_{i=0..n-1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). Also enumerates permutations by their major index.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1, 1, 7, 27, 76, 174, 343, 602, 961, 1415, 1940, 2493, 3017, 3450, 3736, 3836, 3736, 3450, 3017, 2493, 1940, 1415, 961, 602, 343, 174, 76, 27, 7, 1, 1, 8, 35, 111, 285, 628, 1230, 2191, 3606, 5545, 8031, 11021, 14395, 17957, 21450, 24584, 27073, 28675, 29228, 28675, 27073, 24584, 21450, 17957, 14395, 11021, 8031, 5545, 3606, 2191, 1230, 628, 285, 111, 35, 8, 1

Offset: 1

N. J. A. Sloane

T(n,k) is the number of permutations of {1..n} with k inversions.
n-th row gives growth series for symmetric group S_n with respect to transpositions (1,2), (2,3), ..., (n-1,n).
T(n,k) is the number of permutations of (1,2,...,n) having disorder equal to k. The disorder of a permutation p of (1,2,...,n) is defined in the following manner. We scan p from left to right as often as necessary until all its elements are removed in increasing order, scoring one point for each occasion on which an element is passed over and not removed. The disorder of p is the number of points scored by the end of the scanning and removal process. For example, the disorder of (3,5,2,1,4) is 8, since on the first scan, 3,5,2 and 4 are passed over, on the second, 3,5 and 4 and on the third scan, 5 is once again not removed. - Emeric Deutsch, Jun 09 2004
T(n,k) is the number of permutations p=(p(1),...,p(n)) of {1..n} such that Sum_{i: p(i)>p(i+1)} = k (k is called the Major index of p). Example: T(3,0)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1 because the major indices of the permutations (1,2,3), (2,1,3), (3,1,2), (1,3,2), (2,3,1) and (3,2,1) are 0,1,1,2,2 and 3, respectively. - Emeric Deutsch, Aug 17 2004
T(n,k) is the number of 2 X c matrices with column totals 1,2,3,...,n and row totals k and binomial(n+1,2) - k. - Mitch Harris, Jan 13 2006
T(n,k) is the number of permutations p of {1,2,...,n} for which den(p)=k. Here den is the Denert statistic, defined in the following way: let p=p(1)p(2)...p(n) be a permutation of {1,2,...,n}; if p(i)>i, then we say that i is an excedance of p; let i_1 < i_2 < ... < i_k be the excedances of p and let j_1 < j_2 < ... < j_{n-k} be the non-excedances of p; let Exc(p) = p(i_1)p(i_2)...p(i_k), Nexc(p)=p(j_1)p(j_2)...p(j_{n-k}); then, by definition den(p) = i_1 + i_2 + ... + i_k + inv(Exc(p)) + inv(Nexc(p)), where inv denotes "number of inversions". Example: T(4,5)=3 because we have 1342, 3241 and 4321. We show that den(4321)=5: the excedances are 1 and 2; Exc(4321)=43, Nexc(4321)=21; now den(4321) = 1 + 2 + inv(43) + inv(21) = 3+1+1 = 5. - Emeric Deutsch, Oct 29 2008
T(n,k) is the number of size k submultisets of the multiset {1,2,2,3,3,3,...,n-1} (which contains i copies of i for 0 < i < n).
The limit of products of the numbers of fixed necklaces of length n composed of beads of types N(n,b), n --> infinity, is the generating function for inversions (we must exclude one unimportant factor b^n/n!). The error is < (b^n/n!)*O(1/n^(1/2-epsilon)). See Gaichenkov link. - Mikhail Gaichenkov, Aug 27 2012
The number of ways to distribute k-1 indistinguishable balls into n-1 boxes of capacity 1,2,3,...,n-1. - Andrew Woods, Sep 26 2012
Partial sums of rows give triangle A161169. - András Salamon, Feb 16 2013
The number of permutations of n that require k pair swaps in the bubble sort to sort them into the natural 1,2,...,n order. - R. J. Mathar, May 04 2013
Also series coefficients of q-factorial [n]q ! -- see Mathematica line. - _Wouter Meeussen, Jul 12 2014
From Mikhail Gaichenkov, Aug 16 2016: (Start)
Following asymptotic expansions in the Central Limit Theorem developed by Valentin V. Petrov, the cumulative distribution function of these numbers, CDF_N(x), is equal to the CDF of the normal distribution - (0.06/sqrt(2*Pi))*exp(-x^2/2)(x^3-3x)*(6N^3+21N^2+31N+31)/(N(2N+5)^2(N-1)+O(1/N^2).
This can be written as: CDF of the normal distribution -(0.09/(N*sqrt(2*Pi)))*exp(-x^2/2)*He_3(x) + O(1/N^2), N > 1, natural numbers (Gaichenkov, private research).
According to B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4, "the unimodal behavior of the inversion numbers suggests that the number of inversions in a random permutation may be asymptotically normal". See links.
Moreover, E. Ben-Naim (Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory), "On the Mixing of Diffusing Particles" (13 Oct 2010), states that the Mahonian Distribution becomes a function of a single variable for large numbers of element, i.e., the probability distribution function is normal. See links.
To be more precise the expansion of the distribution is presented for a finite number of elements (or particles in terms of E. Ben-Naim's article). The distribution tends to the normal distribution for an infinite numbers of elements.
(End)
T(n,k) statistic counts (labeled) permutation graphs with n vertices and k edges. - Mikhail Gaichenkov, Aug 20 2019
From Gus Wiseman, Aug 12 2020: (Start)
Number of divisors of A006939(n - 1) or A076954(n - 1) with k prime factors, counted with multiplicity, where A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1). For example, row n = 4 counts the following divisors:
  1   2   4   8  24  72 360
      3   6  12  36 120
      5   9  18  40 180
         10  20  60
         15  30  90
             45
Crossrefs:
A336420 is the case with distinct prime multiplicities.
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A317829 counts factorizations of superprimorials.
A336941 counts divisor chains under superprimorials.
Cf. A000005, A001222, A008278, A022559, A027423, A076954, A181796.
(End)
Named after the British mathematician Percy Alexander MacMahon (1854-1929). - Amiram Eldar, Jun 13 2021
Row maxima ~ n!/(sigma * sqrt(2*Pi)), sigma^2 = (2*n^3 + 9*n^2 + 7*n)/72 = variance of group type A_n (see also A161435). - Mikhail Gaichenkov, Feb 08 2023
Sum_{i>=0} T(n,i)*k^i = A069777(n,k). - Geoffrey Critzer, Feb 26 2025

			1; 1+x; (1+x)*(1+x+x^2) = 1+2*x+2*x^2+x^3; etc.
Triangle begins:
  n\k| 0  1   2    3    4     5     6     7     8      9     10
  ---+--------------------------------------------------------------
   1 | 1;
   2 | 1, 1;
   3 | 1, 2,  2,   1;
   4 | 1, 3,  5,   6,   5,    3,    1;
   5 | 1, 4,  9,  15,  20,   22,   20,   15,    9,     4,     1;
   6 | 1, 5, 14,  29,  49,   71,   90,  101,  101,    90,    71, ...
   7 | 1, 6, 20,  49,  98,  169,  259,  359,  455,   531,   573, ...
   8 | 1, 7, 27,  76, 174,  343,  602,  961, 1415,  1940,  2493, ...
   9 | 1, 8, 35, 111, 285,  628, 1230, 2191, 3606,  5545,  8031, ...
  10 | 1, 9, 44, 155, 440, 1068, 2298, 4489, 8095, 13640, 21670, ...
From _Gus Wiseman_, Aug 12 2020: (Start)
Row n = 4 counts the following submultisets of {1,1,1,2,2,3}:
  {}  {1}  {11}  {111}  {1112}  {11122}  {111223}
      {2}  {12}  {112}  {1122}  {11123}
      {3}  {22}  {122}  {1113}  {11223}
           {13}  {113}  {1123}
           {23}  {123}  {1223}
                 {223}
(End)

Miklós Bóna, Combinatorics of permutations, Chapman & Hall/CRC, Boca Raton, Florida, 2004 (p. 52).
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 240.
Florence Nightingale David, Maurice George Kendall, and David Elliot Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
Pierre de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 163, top display.
Eugen Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
Valentin V. Petrov, Sums of Independent Random Variables, Springer Berlin Heidelberg, 1975, p. 134.
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Corollary 1.3.10, p. 21.

Alois P. Heinz, Rows n = 1..50, flattened (first 30 rows from Paul D. Hanna)
Ashleigh Adams, Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, and Amanda Welch, Cyclic sieving on permutations - an analysis of maps and statistics in the FindStat database, arXiv:2402.16251 [math.CO], 2024. See p. 7.
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics, Vol. 35, No. 1 (2019), pp. 1-11.
Hasan Arslan, A combinatorial interpretation of Mahonian numbers of type B, arXiv:2404.05099 [math.CO], 2024. See p. 3.
E. Ben-Naim, On the Mixing of Diffusing Particles, arXiv:1010.2563 [cond-mat.stat-mech], 2010.
David J. Benson, The cohomology of the nilCoxeter algebra, arXiv:2407.21175 [math.RA], 2024. See p. 4.
Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, Tableaux posets and the fake degrees of coinvariant algebras, arXiv:1809.07386 [math.CO], 2018.
Niclas Boehmer, Robert Bredereck, Piotr Faliszewski, and Rolf Niedermeier, On the Robustness of Winners: Counting Briberies in Elections, arXiv:2010.09678 [cs.GT], 2020.
Niclas Boehmer, Robert Bredereck, Piotr Faliszewski, Rolf Niedermeier, and Stanisław Szufa, Putting a Compass on the Map of Elections, arXiv:2105.07815 [cs.GT], 2021.
J. Bourget, Des permutations, Nouvelles annales de mathématiques, 2e série, tome 10 (1871), pp. 254-268.
Franc Brglez, Of n-dimensional Dice, Combinatorial Optimization, and Reproducible Research: An Introduction, Elektrotehniski Vestnik, Vol. 78, No. 4 (2011), pp. 181-192.
Agnieszka Bukietyńska, The test of inversion in the analysis of investment funds, Central and Eastern European Journal of Management and Economics, Vol. 5, No. 3 (September 2017), pp. 277-289.
L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J., Vol. 15, No. 4 (1948), pp. 987-1000.
Luke Chamandy, Anvar Shukurov, and A. Russ Taylor, Statistical tests of galactic dynamo theory, arXiv:1609.05688 [astro-ph.GA], 2016.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Mariusz Czekała and Agnieszka Bukietyńska, Distribution of Inversions and the Power of the τ-Kendall's Test, in J. Świątek, Z. Wilimowska, L. Borzemski, A. Grzech (eds.), Information Systems Architecture and Technology: Proceedings of 37th International Conference on Information Systems Architecture and Technology - ISAT 2016 - Part III, pp. 175-185.
Mariusz Czekała, Agnieszka Bukietyńska, and Agnieszka Matylda Schlichtinger, Estimation of the Probability of Inversion in the Test of Variable Dependencies, Information Systems Architecture and Technology: Proceedings of 39th International Conference on Information Systems Architecture and Technology (ISAT 2018), Part III, 145-156.
F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, pp. 241-242. (Annotated scanned copy)
Emeric Deutsch, Problem 10975: Enumeration of Permutations by Disorder, Amer. Math. Monthly, Vol. 111, No. 6 (2004), p. 541.
FindStat - Combinatorial Statistic Finder, The Denert index of a permutation, The major index of a permutation, The number of inversions of a permutation, The sorting index of a permutation.
Dominique Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, in: M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977, pp. 27-49.
Dominique Foata and Doron Zeilberger, Denert's permutation statistic is indeed Euler-Mahonian, Studies in Appl. Math., Vol. 83 (1990), pp. 31-59. [From _Emeric Deutsch_, Oct 29 2008]
Víctor Franco-Sánchez, Arnau Martí-Llobet, and Ramon Ferrer-i-Cancho, Swap distance minimization beyond entropy minimization in word order variation, arXiv:2404.14192 [cs.CL], 2024. See p. 34.
Mikhail Gaichenkov, Necklaces and the generating function for inversions, MathOverflow, 2012.
Mikhail Gaichenkov, Pros and cons of probability model for permutations, MathOverflow, 2014.
Amy Grady, Sorting index and Mahonian-Stirling pairs for labeled forests, Clemson University, July 10, 2014.
Oğuz Gürerk, Ümit Işlak, and Mehmet Akif Yıldız, A study on random permutation graphs, arXiv:1901.06678 [math.CO], 2019.
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988.
Guo-Niu Han, Une nouvelle bijection pour la statistique de Denert, C. R. Acad. Sci. Paris, Ser. I, Vol. 310 (1990), pp. 493-496.
Stuart A. Hannah, Sieved Enumeration of Interval Orders and Other Fishburn Structures, arXiv:1502.05340 [math.CO], (18-February-2015).
Yang-Hui He, Cyril Matti and Chuang Sun, The Scattering Variety, arXiv preprint arXiv:1403.6833 [hep-th], 2014.
Ekhine Irurozki, Sampling and learning distance-based probability models for permutation spaces, PhD Dissertation, Department of Computer Science and Artificial Intelligence of the University of the Basque Country, 2015.
Ekhine Irurozki, Borja Calvo and José A. Lozano, An R package for permutations, Mallows and Generalized Mallows models, 2014.
Ekhine Irurozki, Borja Calvo and José A. Lozano, PerMallows: An R Package for Mallows and Generalized Mallows Models, Journal of Statistical Software, Vol. 71, No. 12 (August 2016).
Milan Janjic, A Generating Function for Numbers of Insets, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.7.
Thomas Kahle and Christian Stump, Counting inversions and descents of random elements in finite Coxeter groups, arXiv:1802.01389 [math.CO], 2018-2019.
James A. Koziol, Sums of ranking differences and inversion numbers for method discrimination, Journal of Chemometrics, Vol. 27 (2013), pp. 165-169.
Barbara H. Margolius, Permutations with inversions, J. Integ. Seq., Vol. 4 (2001), Article 01.2.4.
Jeremy L. Martin and Jennifer D. Wagner, On the Spectra of Simplicial Rook Graphs, arXiv preprint arXiv:1209.3493 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 27 2012
Anthony Mendes, A note on alternating permutations, Amer. Math. Monthly, Vol. 114, No. 5 (2007), pp. 437-440.
Vinicus de Morais Alves, Rafael Dowsley, Rafael Timateo de Sousa, and Anderson C.A. Nascimento, Information-Theoretically Secure String Commitments Based on Packet Reordering Channels, IEEE Access, Volume 9 (2021), pp. 139931-139936.
Roger H. Moritz and Robert C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., Vol. 61, No. 1 (1988), pp. 24-29.
Eugen Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 92-99 only.
Eugen Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 92-99 only.
Juan Miguel Nieto, Tailoring and Hexagon Form Factors, Spinning Strings and Correlation Functions in the AdS/CFT Correspondence, Springer Theses (Recognizing Outstanding Ph.D. Research), Springer, Cham, 2018.
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Eric Weisstein's World of Mathematics, Necklace.
Eric Weisstein's World of Mathematics, Irreducible Polynomial.
Thomas Wieder, Comments on A008302.
Wikipedia, Major index

Diagonals: A000707 (k=n-1), A001892 (k=n-2), A001893 (k=n-3), A001894 (k=n-4), A005283 (k=n-5), A005284 (k=n-6), A005285 (k=n-7).
Columns: A005286 (k=3), A005287 (k=4), A005288 (k=5), A242656 (k=6), A242657 (k=7).
Rows: A161435 (n=4), A161436 (n=5), A161437 (n=6), A161438 (n=7), A161439 (n=8), A161456 (n=9), A161457 (n=10).
Row-maxima: A000140, truncated table: A060701, row sums: A000142, row lengths: A000124.
A001809 gives total Denert index of all permutations.
A357611 gives a refinement.
Cf. A000217, A139365, A069777.

g := proc(n,k) option remember; if k=0 then return(1) else if (n=1 and k=1) then return(0) else if (k<0 or k>binomial(n,2)) then return(0) else g(n-1,k)+g(n,k-1)-g(n-1,k-n) end if end if end if end proc; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001
BB:=j->1+sum(t^i, i=1..j): for n from 1 to 8 do Z[n]:=sort(expand(simplify(product(BB(j), j=0..n-2)))) od: for n from 1 to 8 do seq(coeff(Z[n], t, j), j=0..(n-1)*(n-2)/2) od; # Zerinvary Lajos, Apr 13 2007
# alternative Maple program:
b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(u+j-1, o-j)*x^(u+j-1), j=1..o)+
       add(b(u-j, o+j-1)*x^(u-j), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 02 2017

f[n_] := CoefficientList[ Expand@ Product[ Sum[x^i, {i, 0, j}], {j, n}], x]; Flatten[Array[f, 8, 0]]
(* Second program: *)
T[0, 0] := 1; T[-1, k_] := 0;
T[n_, k_] := T[n, k] = If[0 <= k <= n*(n - 1)/2, T[n, k - 1] + T[n - 1, k] - T[n - 1, k - n], 0]; (* Peter Kagey, Mar 18 2021; corrected the program by Mats Granvik and Roger L. Bagula, Jun 19 2011 *)
alternatively (versions 7 and up):
Table[CoefficientList[Series[QFactorial[n,q],{q,0,n(n-1)/2}],q],{n,9}] (* Wouter Meeussen, Jul 12 2014 *)
b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
   Sum[b[u + j - 1, o - j]*x^(u + j - 1), {j, 1, o}] +
   Sum[b[u - j, o + j - 1]*x^(u - j), {j, 1, u}]]];
T[n_] := With[{p = b[n, 0]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 21 2025, after Alois P. Heinz *)

{T(n,k) = my(A=1+x); for(i=1,n, A = 1 + intformal(A - q*subst(A,x,q*x +x^2*O(x^n)))/(1-q)); polcoeff(n!*polcoeff(A,n,x),k,q)}
for(n=1,10, for(k=0,n*(n-1)/2, print1(T(n,k),", ")); print("")) \\ Paul D. Hanna, Dec 31 2016

row(n)=Vec(prod(k=1,n,(1-'q^k)/(1-'q))); \\ Joerg Arndt, Apr 13 2019

from sage.combinat.q_analogues import q_factorial
for n in (1..6): print(q_factorial(n).list()) # Peter Luschny, Jul 18 2016

Bourget, Comtet and Moritz-Williams give recurrences.
Mendes and Stanley give g.f.'s.
G.f.: Product_{j=1..n} (1-x^j)/(1-x) = Sum_{k=0..M} T{n, k} x^k, where M = n*(n-1)/2.
From Andrew Woods, Sep 26 2012, corrected by Peter Kagey, Mar 18 2021: (Start)
T(1, 0) = 1,
T(n, k) = 0 for n < 0, k < 0 or k > n*(n-1)/2.
T(n, k) = Sum_{j=0..n-1} T(n-1, k-j),
T(n, k) = T(n, k-1) + T(n-1, k) - T(n-1, k-n). (End)
E.g.f. satisfies: A(x,q) = 1 + Integral (A(x,q) - q*A(q*x,q))/(1-q) dx, where A(x,q) = Sum_{n>=0} x^n/n! * Sum_{k=0..n*(n-1)/2} T(n,k)*q^k, when T(0,0) = 1 is included. - Paul D. Hanna, Dec 31 2016

There were some mistaken edits to this entry (inclusion of an initial 1, etc.) which I undid. - N. J. A. Sloane, Nov 30 2009

Added mention of "major index" to definition. - N. J. A. Sloane, Feb 10 2019

cp's OEIS Frontend

A179998 Numbers which are members of both A000140 and A039909.

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A186888 Indices of odd Kendall-Mann numbers (A000140).

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A173489 Partial sums of A000140.

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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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A000124 Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.

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A008302 Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_{i=0..n-1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). Also enumerates permutations by their major index.

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

A186772 Largest coefficient of (1)(1+1x)(1+1x+2x^2)...(1+1x+2x^2+...+n*x^n).