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