A010790 a(n) = n!*(n+1)!.

1, 2, 12, 144, 2880, 86400, 3628800, 203212800, 14631321600, 1316818944000, 144850083840000, 19120211066880000, 2982752926433280000, 542861032610856960000, 114000816848279961600000, 27360196043587190784000000, 7441973323855715893248000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Let M_n be the symmetrical n X n matrix M_n(i,j)=1/min(i,j); then for n>=0 det(M_n)=(-1)^(n-1)/a(n-1). - Benoit Cloitre, Apr 27 2002

If n women and n men are to be seated around a circular table, with no two of the same sex seated next to each other, the number of possible arrangements is a(n-1). - Ross La Haye, Jan 06 2009

a(n-1) is also the number of (directed) Hamiltonian cycles in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 15 2011

a(n) is also number of ways to place k nonattacking semi-bishops on an n X n board, sum over all k>=0 (for definition see A187235). - Vaclav Kotesovec, Dec 06 2011

a(n) is number of permutations of {1,2,3,...,2n} such that no odd numbers are adjacent. - Ran Pan, May 23 2015

a(n) is number of permutations of {1,2,3,...,2n+1} such that no odd numbers are adjacent. - Ran Pan, May 23 2015

a(n-1) is the number of elements of the wreath product of S_n and S_2 with cycle partition equal to (2n), where S_n is the symmetric group of order n. - Josaphat Baolahy, Mar 12 2024

Examples

			G.f. = 1 + 2*x + 12*x^2 + 144*x^3 + 2880*x^4 + 86400*x^5 + ...
		

References

• J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 63-65.
• Kenneth H. Rosen, Editor-in-Chief, Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000, page 91. [Ross La Haye, Jan 06 2009]

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
• Steve Gadbois, 104.12 From calendar coincidence to factorials to Ramanujan, The Mathematical Gazette (2020) Vol. 104, Issue 560, 304-306.
• Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
• Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 268.
• S. Tanimoto, Parity alternating permutations and signed Eulerian numbers, Ann. Comb. 14 (2010) 355 (total number of PAPs of [2n+1].)
• Eric Weisstein's World of Mathematics, Complete Bipartite Graph
• Eric Weisstein's World of Mathematics, Hamiltonian Cycle
• Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).
• Index entries for sequences related to factorial numbers

Crossrefs

Second column of triangle A129065.

Cf. A004737, A000290, A000108, A126120.

Programs

Magma

[Factorial(n)*Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014

Maple

f:= n-> n!*(n+1)!: seq(f(n), n=0..30);

Mathematica

s=1;lst={s};Do[s+=(s*=n)*n;AppendTo[lst, s], {n, 1, 4!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Times@@@Partition[Range[0,25]!,2,1] (* Harvey P. Dale, Jun 17 2011 *)

PARI

a(n)= n!^2*(n+1) \\ Charles R Greathouse IV, Jul 31 2011

Python

from math import factorial
def A010790(n): return factorial(n)**2*(n+1) # Chai Wah Wu, Apr 22 2024

Sage

[stirling_number1(n,1)*factorial (n-2) for n in range(2, 17)] # Zerinvary Lajos, Jul 07 2009

Formula

From Karol A. Penson, Oct 23 2001: (Start)

Integral representation as n-th moment of a positive function f on the positive half axis, where f(x) = 2*sqrt(x)*BesselK(1, 2*sqrt(x)). Then:

a(n) = Integral_{x>=0} x^n * f(x) dx.

G.f.: a(0) = 1 and a(n) = subs(x=0, n!*diff(1/((x-1)^2), x$n)) for n >= 1. (End)

Sum_{i >=0} 1/a(i) = A096789. - Gerald McGarvey, Jun 10 2004

With b(n)=A002378(n) for n>0 and b(0)=1, a(n) = b(n)*b(n-1)...*b(0). - Tom Copeland, Sep 21 2011

a(n) = det(PS(i+1,j), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013

a(n) = (2*n)! / A000108(n) which implies that the e.g.f. of A126120 is Sum_{k>=0} x^(2*k) / a(k). - Michael Somos, Nov 15 2014

0 = a(n)*(+18*a(n+2) - 15*a(n+3) + a(n+4)) + a(n+1)*(-9*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) for all n>=0. - Michael Somos, Nov 15 2014

From Ilya Gutkovskiy, Jan 20 2017: (Start)

a(n) ~ 2*Pi*n^(2*n+2)/exp(2*n).

Sum_{n>=0} (-1)^n/a(n) = BesselJ(1,2) = 0.576724807756873387202448... = A348607 (End)

D-finite with recurrence: a(n) -n*(n+1)*a(n-1)=0. - R. J. Mathar, Jan 27 2020

a(n) = 1/([x^n] hypergeom([], [2], x)). - Peter Luschny, Sep 13 2024