A277392 a(n) = n!*LaguerreL(n, -3*n).

1, 4, 62, 1626, 59928, 2844120, 165100752, 11331597942, 897635712384, 80602042275756, 8090067511468800, 897561658361441106, 109072492644378442752, 14407931244544181001216, 2055559499598438969956352, 314997663481165477898736750, 51601245736595962597616222208

Offset: 0

Author

Vaclav Kotesovec, Oct 12 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. A002720, A087912, A277382.

Cf. A277373, A277391, A277418, A277419, A277420, A277421, A277422.

Programs

Magma

[Factorial(n)*(&+[Binomial(n,k)*3^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Mathematica

Table[n!*LaguerreL[n, -3*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k]*3^k*n^k/k!, {k, 0, n}], {n, 1, 20}]}]

PARI

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*3^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

Formula

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 3^k * n^k / k!.

a(n) ~ sqrt(1/2+5/(2*sqrt(21))) * (5+sqrt(21))^n * exp(n*(sqrt(21)-5)/2) * n^n/2^n.