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

%I A010790 #120 Feb 16 2025 08:32:32
%S A010790 1,2,12,144,2880,86400,3628800,203212800,14631321600,1316818944000,
%T A010790 144850083840000,19120211066880000,2982752926433280000,
%U A010790 542861032610856960000,114000816848279961600000,27360196043587190784000000,7441973323855715893248000000
%N A010790 a(n) = n!*(n+1)!.
%C A010790 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 A010790 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 A010790 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 A010790 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 A010790 a(n) is number of permutations of {1,2,3,...,2n} such that no odd numbers are adjacent. - _Ran Pan_, May 23 2015
%C A010790 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 A010790 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 A010790 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 63-65.
%D A010790 Kenneth H. Rosen, Editor-in-Chief, Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000, page 91. [_Ross La Haye_, Jan 06 2009]
%H A010790 T. D. Noe, <a href="/A010790/b010790.txt">Table of n, a(n) for n = 0..100</a>
%H A010790 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 A010790 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 A010790 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 A010790 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 268.
%H A010790 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 A010790 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A010790 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>
%H A010790 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 A010790 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A010790 From _Karol A. Penson_, Oct 23 2001: (Start)
%F A010790 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 A010790 a(n) = Integral_{x>=0} x^n * f(x) dx.
%F A010790 G.f.: a(0) = 1 and a(n) = subs(x=0, n!*diff(1/((x-1)^2), x$n)) for n >= 1. (End)
%F A010790 Sum_{i >=0} 1/a(i) = A096789. - _Gerald McGarvey_, Jun 10 2004
%F A010790 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 A010790 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 A010790 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 A010790 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 A010790 From _Ilya Gutkovskiy_, Jan 20 2017: (Start)
%F A010790 a(n) ~ 2*Pi*n^(2*n+2)/exp(2*n).
%F A010790 Sum_{n>=0} (-1)^n/a(n) = BesselJ(1,2) = 0.576724807756873387202448... = A348607 (End)
%F A010790 D-finite with recurrence: a(n) -n*(n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 27 2020
%F A010790 a(n) = 1/([x^n] hypergeom([], [2], x)). - _Peter Luschny_, Sep 13 2024
%e A010790 G.f. = 1 + 2*x + 12*x^2 + 144*x^3 + 2880*x^4 + 86400*x^5 + ...
%p A010790 f:= n-> n!*(n+1)!: seq(f(n), n=0..30);
%t A010790 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 A010790 Times@@@Partition[Range[0,25]!,2,1] (* _Harvey P. Dale_, Jun 17 2011 *)
%o A010790 (Sage) [stirling_number1(n,1)*factorial (n-2) for n in range(2, 17)] # _Zerinvary Lajos_, Jul 07 2009
%o A010790 (PARI) a(n)= n!^2*(n+1) \\ _Charles R Greathouse IV_, Jul 31 2011
%o A010790 (Magma) [Factorial(n)*Factorial(n+1): n in [0..20]]; // _Vincenzo Librandi_, Aug 08 2014
%o A010790 (Python)
%o A010790 from math import factorial
%o A010790 def A010790(n): return factorial(n)**2*(n+1) # _Chai Wah Wu_, Apr 22 2024
%Y A010790 Second column of triangle A129065.
%Y A010790 Cf. A004737, A000290, A000108, A126120.
%K A010790 nonn,nice,easy
%O A010790 0,2
%A A010790 _N. J. A. Sloane_