A010790 - OEIS

%I #119 Sep 13 2024 11:29:40

%S 1,2,12,144,2880,86400,3628800,203212800,14631321600,1316818944000,

%T 144850083840000,19120211066880000,2982752926433280000,

%U 542861032610856960000,114000816848279961600000,27360196043587190784000000,7441973323855715893248000000

%N a(n) = n!*(n+1)!.

%C 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

%C 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

%C 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

%C 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

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

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

%C 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

%D J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 63-65.

%D Kenneth H. Rosen, Editor-in-Chief, Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000, page 91. [_Ross La Haye_, Jan 06 2009]

%H T. D. Noe, <a href="/A010790/b010790.txt">Table of n, a(n) for n = 0..100</a>

%H J. Agapito, <a href="http://dx.doi.org/10.1016/j.laa.2014.03.018">On symmetric polynomials with only real zeros and nonnegative gamma-vectors</a>, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.

%H Steve Gadbois, <a href="https://doi.org/10.1017/mag.2020.54">104.12 From calendar coincidence to factorials to Ramanujan</a>, The Mathematical Gazette (2020) Vol. 104, Issue 560, 304-306.

%H Anatol N. Kirillov, <a href="https://doi.org/10.3842/SIGMA.2016.002">On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials</a>, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 268.

%H S. Tanimoto, <a href="http://dx.doi.org/10.1007/s00026-010-0064-3">Parity alternating permutations and signed Eulerian numbers</a>, Ann. Comb. 14 (2010) 355 (total number of PAPs of [2n+1].)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>

%H Shawn L. Witte, <a href="https://www.math.ucdavis.edu/~tdenena/dissertations/201910_Witte_Dissertation.pdf">Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory</a>, Ph. D. Dissertation, University of California-Davis (2020).

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F From _Karol A. Penson_, Oct 23 2001: (Start)

%F 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:

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

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

%F Sum_{i >=0} 1/a(i) = A096789. - _Gerald McGarvey_, Jun 10 2004

%F 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

%F 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

%F 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

%F 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

%F From _Ilya Gutkovskiy_, Jan 20 2017: (Start)

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

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

%F D-finite with recurrence: a(n) -n*(n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 27 2020

%F a(n) = 1/([x^n] hypergeom([], [2], x)). - _Peter Luschny_, Sep 13 2024

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

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

%t s=1;lst={s};Do[s+=(s*=n)*n;AppendTo[lst, s], {n, 1, 4!, 1}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 15 2008 *)

%t Times@@@Partition[Range[0,25]!,2,1] (* _Harvey P. Dale_, Jun 17 2011 *)

%o (Sage) [stirling_number1(n,1)*factorial (n-2) for n in range(2, 17)] # _Zerinvary Lajos_, Jul 07 2009

%o (PARI) a(n)= n!^2*(n+1) \\ _Charles R Greathouse IV_, Jul 31 2011

%o (Magma) [Factorial(n)*Factorial(n+1): n in [0..20]]; // _Vincenzo Librandi_, Aug 08 2014

%o (Python)

%o from math import factorial

%o def A010790(n): return factorial(n)**2*(n+1) # _Chai Wah Wu_, Apr 22 2024

%Y Second column of triangle A129065.

%Y Cf. A004737, A000290, A000108, A126120.

%K nonn,nice,easy

%O 0,2

%A _N. J. A. Sloane_