A277391 a(n) = n!*LaguerreL(n, -2*n).

1, 3, 34, 654, 17688, 616120, 26252496, 1322624016, 76909665664, 5069558461824, 373529452588800, 30422117430022912, 2713911389090970624, 263171888496899625984, 27563036166079327578112, 3100736138961250867968000, 372888702864658105915244544

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, A277392, A277418, A277419, A277420, A277421, A277422.

Programs

Magma

[Factorial(n)*(&+[Binomial(n,k)*2^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Mathematica

Table[n!*LaguerreL[n, -2*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k]*2^k*n^k/k!, {k, 0, n}], {n, 1, 20}]}]

PARI

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*2^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

Formula

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 2^k * n^k / k!.

a(n) ~ (1 + sqrt(3))^(2*n+1) * n^n / (3^(1/4) * 2^(n+1) * exp((2 - sqrt(3))*n)).