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A295408 a(n) = n! * Laguerre(n, 4*n, -n). 7
1, 6, 134, 5052, 267576, 18246850, 1521907056, 150077897088, 17080661438336, 2203559337858174, 317761804144896000, 50650336389453807556, 8843008543955452118016, 1678231571506037926192698, 343989152383931539269349376, 75733086648535784012234565000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

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

FORMULA

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

MATHEMATICA

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}]]

PROG

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

(MAGMA) [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

CROSSREFS

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

Cf. A295409, A295418, A332679.

Sequence in context: A244745 A179564 A263583 * A301463 A220064 A050281

Adjacent sequences:  A295405 A295406 A295407 * A295409 A295410 A295411

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Nov 22 2017

STATUS

approved

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Last modified January 26 10:45 EST 2021. Contains 340438 sequences. (Running on oeis4.)