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A224900
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a(n) = n!*((n+1)!)^2.
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3
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1, 4, 72, 3456, 345600, 62208000, 18289152000, 8193540096000, 5309413982208000, 4778472583987200000, 5781951826624512000000, 9158611693373227008000000, 18573664514160904372224000000, 47325697182081984340426752000000, 149075946123558250672344268800000000
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OFFSET
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0,2
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COMMENTS
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2*a(n-1) is the number of elements of the wreath product of S_n and S_3 with cycle partition equal to (3n). - Josaphat Baolahy, Mar 12 2024
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LINKS
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FORMULA
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G.f. of hypergeometric type: Sum_{n>=0} a(n)*z^n/(n!)^3 = (1+z)/(1-z)^3.
Integral representation as n-th moment of a positive function w(x) on a positive half axis (solution of the Stieltjes moment problem), in Maple notation: a(n) = int(x^n*w(x),x=0..infinity), n>=0, where w(x) = MeijerG([[],[]],[[1,1,0]],[]],x), w(0)=1, limit(w(x),x=infinity)=0.
w(x) is monotonically decreasing over (0,infinity).
The Meijer G function above cannot be represented by any other known special function.
This solution of the Stieltjes moment problem is not unique.
Asymptotics: a(n) -> (1/960)*sqrt(2)*Pi^(3/2)*(1920*n^3 + 4320*n^2 + 2940*n + 589)*exp(-3*n)*n^(1/2 + 3*n), for n->oo.
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MATHEMATICA
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Table[n!*((n+1)!)^2, {n, 0, 15}]
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PROG
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(Python)
from math import factorial
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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