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A295408 a(n) = n! * Laguerre(n, 4*n, -n). 7

%I #24 Sep 08 2022 08:46:20

%S 1,6,134,5052,267576,18246850,1521907056,150077897088,17080661438336,

%T 2203559337858174,317761804144896000,50650336389453807556,

%U 8843008543955452118016,1678231571506037926192698,343989152383931539269349376,75733086648535784012234565000

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

%C 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.

%H G. C. Greubel, <a href="/A295408/b295408.txt">Table of n, a(n) for n = 0..300</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

%F a(n) = n!*Sum_{k=0..n} binomial(5*n,n-k)*n^k/k!.

%F a(n) ~ sqrt(1/2 + 7/(2*sqrt(29))) * (131 - 22*sqrt(29))^n * exp((sqrt(29)-7)*n/2) * n^n.

%F a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(4*n+1). - _Ilya Gutkovskiy_, Nov 23 2017

%t Table[n!*LaguerreL[n,4*n,-n],{n,0,15}]

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

%o (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

%o (PARI) a(n) = n!*pollaguerre(n, 4*n, -n); \\ _Michel Marcus_, Feb 05 2021

%o (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

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

%Y Cf. A295409, A295418, A332679.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 22 2017

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)