A277419 a(n) = n!*LaguerreL(n, -5*n).

1, 6, 142, 5676, 318744, 23046370, 2038090320, 213094791840, 25714702990720, 3517403388684030, 537798502938028800, 90890936781714193300, 16825134146527678233600, 3385560150770468257273050, 735772370353606135149107200, 171753027520961356975091493000

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, A277418, A277420, A277421, A277422.

Cf. A002720, A087912, A277382.

Programs

Magma

[Factorial(n)*(&+[Binomial(n,k)*5^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Mathematica

Table[n!*LaguerreL[n, -5*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * 5^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]

PARI

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*5^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

Formula

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

a(n) ~ sqrt(1/2 + 7/(6*sqrt(5))) * ((7 + 3*sqrt(5))/2)^n * exp((-7 + 3*sqrt(5))*n/2) * n^n.

Equivalently, a(n) ~ phi^(4*n + 2) * n^n / (sqrt(3) * 5^(1/4) * exp(n/phi^4)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021