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A234971
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a(n) = Sum_{k=0..n} n^k * binomial(n,k)^4.
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4
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1, 2, 37, 1000, 38401, 1896876, 112124629, 7679202336, 595411451905, 51348552829300, 4861414171762501, 500163335120177136, 55466421261812540929, 6585829687114412247800, 832587068884779776276661, 111541424966889778569909376, 15771414153994526723881828353
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OFFSET
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0,2
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COMMENTS
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In general, Sum_{k=0..n} n^k * binomial(n,k)^p is asymptotic to (1+n^(1/p))^(n*p+p-1) / sqrt(p * (2*Pi)^(p-1) * n^(p-1/p)).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200 [a(0)=1 inserted by Georg Fischer, Jan 04 2020]
Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012
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FORMULA
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a(n) ~ (1+n^(1/4))^(4*n+3) / (4*sqrt(2) * Pi^(3/2) * n^(15/8)).
a(n) = hypergeom([-n, -n, -n, -n], [1, 1, 1], n). - Peter Luschny, Dec 22 2020
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MAPLE
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a := n -> hypergeom([-n, -n, -n, -n], [1, 1, 1], n):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Dec 22 2020
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MATHEMATICA
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Table[Sum[If[n==k==0, 1, n^k]*Binomial[n, k]^4, {k, 0, n}], {n, 0, 20}] (* offset adapted by Georg Fischer, Jan 04 2021 *)
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PROG
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(PARI) a(n) = sum(k=0, n, n^k * binomial(n, k)^4); \\ Michel Marcus, Jan 04 2021
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CROSSREFS
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Cf. A187021, A241247.
Sequence in context: A307318 A058245 A257995 * A139108 A165697 A320994
Adjacent sequences: A234968 A234969 A234970 * A234972 A234973 A234974
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KEYWORD
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nonn,easy
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AUTHOR
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Vaclav Kotesovec, Apr 19 2014
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EXTENSIONS
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a(0) = 1 prepended by Peter Luschny, Dec 22 2020
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STATUS
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approved
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