A277418 a(n) = n!*LaguerreL(n, -4*n).

1, 5, 98, 3246, 151064, 9052120, 663449040, 57490690544, 5749754436992, 651830574374784, 82599621627948800, 11569798584488362240, 1775052172071446510592, 296026752508667034942464, 53320241823337034415908864, 10315767337287172256717568000

Offset: 0

Author

Vaclav Kotesovec, Oct 14 2016

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials

Crossrefs

Cf. A277373, A277391, A277392, A277419, A277420, A277421, A277422.

Cf. A002720, A087912, A277382, A332679.

Programs

Magma

[Factorial(n)*(&+[Binomial(n,k)*4^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Mathematica

Table[n!*LaguerreL[n, -4*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * 4^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]

PARI

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*4^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

Formula

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 4^k * n^k / k!.

a(n) ~ sqrt(2 + 3/sqrt(2)) * (3 + 2*sqrt(2))^n * exp((-3 + 2*sqrt(2))*n) * n^n / 2.