OFFSET
0,2
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
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).
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
EXAMPLE
G.f. = 1 + 2*x + 12*x^2 + 144*x^3 + 2880*x^4 + 86400*x^5 + ...
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 *)
PROG
(Sage) [stirling_number1(n, 1)*factorial (n-2) for n in range(2, 17)] # Zerinvary Lajos, Jul 07 2009
(PARI) a(n)= n!^2*(n+1) \\ Charles R Greathouse IV, Jul 31 2011
(Magma) [Factorial(n)*Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014
(Python)
from math import factorial
def A010790(n): return factorial(n)**2*(n+1) # Chai Wah Wu, Apr 22 2024
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved