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A010790
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a(n) = n!*(n+1)!.
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49
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1, 2, 12, 144, 2880, 86400, 3628800, 203212800, 14631321600, 1316818944000, 144850083840000, 19120211066880000, 2982752926433280000, 542861032610856960000, 114000816848279961600000, 27360196043587190784000000, 7441973323855715893248000000
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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]
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LINKS
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FORMULA
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Integral representation as n-th moment of a positive function on a positive half axis, in Maple notation: a(n)=int(x^n*2*sqrt(x)*BesselK(1, 2*sqrt(x)), x=0..infinity), n=0, 1... Hypergeometric g.f.: a(0)=1, a(n)=subs(x=0, n!*diff(1/((x-1)^2), x$n)), n=1, 2... - Karol A. Penson, Oct 23 2001
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
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
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
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EXAMPLE
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G.f. = 1 + 2*x + 12*x^2 + 144*x^3 + 2880*x^4 + 86400*x^5 + ...
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MAPLE
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f:= n-> n!*(n+1)!: seq(f(n), n=0..30);
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MATHEMATICA
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Times@@@Partition[Range[0, 25]!, 2, 1] (* Harvey P. Dale, Jun 17 2011 *)
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PROG
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(Sage) [stirling_number1(n, 1)*factorial (n-2) for n in range(2, 17)] # Zerinvary Lajos, Jul 07 2009
(Magma) [Factorial(n)*Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014
(Python)
from math import factorial
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CROSSREFS
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KEYWORD
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nonn,nice,easy,changed
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AUTHOR
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STATUS
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approved
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