A277382 a(n) = n!*LaguerreL(n, -3).

1, 4, 23, 168, 1473, 14988, 173007, 2228544, 31636449, 490102164, 8219695239, 148262469336, 2860241078817, 58736954622492, 1278727896354687, 29406849577341552, 712119108949808193, 18108134430393657636, 482306685868464422391, 13425231879291031821576

Offset

0,2

Comments

For m > 0, n!*LaguerreL(n, -m) ~ exp(2*sqrt(m*n) - n - m/2) * n^(n + 1/4) / (sqrt(2)*m^(1/4)) * (1 + (3+24*m+4*m^2)/(48*sqrt(m*n))).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..438

W. Van Assche, Erratum to "Weighted zero distribution for polynomials orthogonal on an infinite interval", SIAM J. Math. Anal., 32 (2001), 1169-1170.

Oskar Perron, Über das Verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem Wachstum eines Parameters, Journal für die reine und angewandte Mathematik (1921), vol. 151, p. 63-78.

Eric Weisstein's World of Mathematics, Laguerre Polynomial

Wikipedia, Laguerre polynomials

Index entries for sequences related to Laguerre polynomials

Formula

E.g.f.: exp(3*x/(1-x))/(1-x).

a(n) = Sum_{k=0..n} 3^k*(n-k)!*binomial(n, k)^2.

a(n) ~ exp(2*sqrt(3*n)-n-3/2) * n^(n+1/4) / (sqrt(2) * 3^(1/4)) * (1 + 37/(16*sqrt(3*n))).

D-finite with recurrence a(n) = 2*(n+1)*a(n-1) - (n-1)^2*a(n-2).

Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(3*n))) = exp(-3/2).

a(n) = n! * A160613(n)/A160614(n). - Alois P. Heinz, Jun 28 2017

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 3^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

Mathematica

Table[n!*LaguerreL[n, -3], {n, 0, 20}]
CoefficientList[Series[E^(3*x/(1-x))/(1-x), {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[Binomial[n, k]^2 * 3^k * (n-k)!, {k, 0, n}], {n, 0, 20}]

Prog

(PARI) for(n=0, 30, print1(n!*(sum(k=0, n, binomial(n, k)*(3^k/k!))), ", ")) \\ G. C. Greubel, May 09 2018
(Magma) [Factorial(n)*((&+[Binomial(n, k)*(3^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 09 2018

Crossrefs

Cf. A002720, A087912, A277372.

Column k=3 of A289192.

Cf. A160613, A160614.

Sequence in context: A158884 A317057 A053525 * A208676 A317276 A113869

Adjacent sequences: A277379 A277380 A277381 * A277383 A277384 A277385

Keyword

nonn

Author

Vaclav Kotesovec, Oct 12 2016

Status

approved