A277422 a(n) = n!*LaguerreL(n, -8*n).

1, 9, 322, 19446, 1649688, 180184120, 24070390992, 3801662863152, 692979602529664, 143184960501077376, 33069665092749868800, 8442378658666161822976, 2360674573114695421197312, 717531421372546588398529536, 235551703250624390582942574592

Offset: 0

Author

Vaclav Kotesovec, Oct 14 2016

Keywords

• nonn

Comments

In general, if m > 0 and a(n) = n!*LaguerreL(n, -m*n), then a(n) ~ sqrt(1/2 + (m+2)/(2*sqrt(m*(m+4)))) * (2+m+sqrt(m*(m+4)))^n * exp(n*(sqrt(m*(m+4))-m-2)/2) * n^n / 2^n.

For m > 4, (-1)^n * n! * LaguerreL(n, m*n) ~ sqrt(1/2 + (m-2)/(2*sqrt(m*(m-4)))) * exp((m - 2 - sqrt(m*(m-4)))*n/2) * ((m - 2 + sqrt(m*(m-4)))/2)^n * n^n. - Vaclav Kotesovec, Feb 20 2020

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 (m=1), A277391 (m=2), A277392 (m=3), A277418 (m=4), A277419 (m=5), A277420 (m=6), A277421 (m=7).

Cf. A002720, A087912, A277382.

Programs

Magma

[Factorial(n)*(&+[Binomial(n,k)*(8)^k*n^k/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, May 16 2018

Mathematica

Table[n!*LaguerreL[n, -8*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * 8^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]

PARI

for(n=0, 30, print1(n!*sum(k=0,n, binomial(n,k)*(8)^k*n^k/k!), ", ")) \\ G. C. Greubel, May 16 2018

Formula

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 8^k * n^k / k!.

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