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A295409
a(n) = n! * Laguerre(n, n^2, -n).
4
1, 3, 58, 2859, 267576, 40818095, 9235507968, 2906955312471, 1215257338052992, 651548571287972859, 435901423022852332800, 356000439852418418920643, 348583395952381998326141952, 403108990190536860168604229031, 543577365164816368801494214352896
OFFSET
0,2
FORMULA
a(n) = n! * Sum_{k=0..n} binomial(n*(n+1),n-k)*n^k/k!.
a(n) ~ exp(3/2) * n^(2*n).
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Nov 23 2017
MAPLE
seq(n!*orthopoly[L](n, n^2, -n), n=0..30); # Robert Israel, Nov 22 2017
MATHEMATICA
Table[n!*LaguerreL[n, n^2, -n], {n, 0, 15}]
Join[{1}, Table[n!*Sum[Binomial[n*(n+1), n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]
PROG
(PARI) for(n=0, 30, print1(n!*sum(k=0, 30, binomial(n*(n+1), n-k)*n^k/k!), ", ")) \\ G. C. Greubel, May 11 2018
(PARI) a(n) = n!*pollaguerre(n, n^2, -n); \\ Michel Marcus, Feb 05 2021
(Magma) [Factorial(n)*(&+[Binomial(n*(n+1), n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 11 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Nov 22 2017
STATUS
approved