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
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).
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
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 12 2016
STATUS
approved