A295418 -id:A295418 - cp's OEIS frontend

A295408 a(n) = n! * Laguerre(n, 4*n, -n).

1, 6, 134, 5052, 267576, 18246850, 1521907056, 150077897088, 17080661438336, 2203559337858174, 317761804144896000, 50650336389453807556, 8843008543955452118016, 1678231571506037926192698, 343989152383931539269349376, 75733086648535784012234565000

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

In general, for fixed m >= 1, n! * Sum_{k=0..n} binomial(m*n, n-k) * n^k / k! = n! * Laguerre(n, (m-1)*n, -n) ~ sqrt(1/2 + (m + 2)/(2*sqrt(m^2 + 4))) * (2^(m+1) * m^m / ((sqrt(m^2 + 4) - m) * (m - 2 + sqrt(m^2 + 4))^m))^n * exp((sqrt(m^2 + 4) - m)*n/2 - n) * n^n.

G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373 (m=1), A295385 (m=2), A295406 (m=3), A295407 (m=4).

Cf. A295409, A295418, A332679.

[Factorial(n)*(&+[Binomial(5*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n!*LaguerreL[n,4*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[5*n,n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

for(n=0,30, print1(n!*sum(k=0, n, binomial(5*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

a(n) = n!*pollaguerre(n, 4*n, -n); \\ Michel Marcus, Feb 05 2021

a(n) = n!*Sum_{k=0..n} binomial(5*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 7/(2*sqrt(29))) * (131 - 22*sqrt(29))^n * exp((sqrt(29)-7)*n/2) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(4*n+1). - Ilya Gutkovskiy, Nov 23 2017

A295409 a(n) = n! * Laguerre(n, n^2, -n).

Original entry on oeis.org

1, 3, 58, 2859, 267576, 40818095, 9235507968, 2906955312471, 1215257338052992, 651548571287972859, 435901423022852332800, 356000439852418418920643, 348583395952381998326141952, 403108990190536860168604229031, 543577365164816368801494214352896

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..214
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295406, A295407, A295408, A295418.

[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

seq(n!*orthopoly[L](n,n^2,-n),n=0..30); # Robert Israel, Nov 22 2017

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}]]

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

a(n) = n!*pollaguerre(n, n^2, -n); \\ Michel Marcus, Feb 05 2021

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

cp's OEIS Frontend

A295408 a(n) = n! * Laguerre(n, 4*n, -n).

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A295409 a(n) = n! * Laguerre(n, n^2, -n).

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Magma

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Mathematica

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