OFFSET
0,2
COMMENTS
In general, if e.g.f. = exp(Sum_{k>=1} m*x^k) = exp(m*x/(1-x)) and m>0, then a(n) ~ n! * m^(1/4) * exp(2*sqrt(m*n) - m/2) / (2 * sqrt(Pi) * n^(3/4)).
LINKS
FORMULA
E.g.f.: exp(3*x/(1-x)).
a(n) ~ 3^(1/4) * exp(2*sqrt(3*n) - 3/2) * n! / (2*sqrt(Pi)*n^(3/4)).
a(n) = (2*n+1)*a(n-1) - (n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Nov 04 2016
From G. C. Greubel, Feb 24 2021: (Start)
a(n) = A253286(n+3, 3).
a(n) = 3*(n-1)!*LaguerreL(n-1, 1, -3) with a(0) = 1. (End)
MATHEMATICA
nmax=20; CoefficientList[Series[Exp[Sum[3*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
CoefficientList[Series[E^(3*x/(1-x)), {x, 0, 20}], x] * Range[0, 20]!
Table[If[n==0, 1, 3*(n-1)!*LaguerreL[n-1, 1, -3]], {n, 0, 25}] (* G. C. Greubel, Feb 24 2021 *)
PROG
(PARI) my(x='x +O('x^50)); Vec(serlaplace(exp(3*x/(1-x)))) \\ G. C. Greubel, Feb 05 2017
(Sage) [1 if n==0 else 3*factorial(n-1)*gen_laguerre(n-1, 1, -3) for n in (0..25)] # G. C. Greubel, Feb 24 2021
(Magma) [n eq 0 select 1 else 3*Factorial(n-1)*Evaluate(LaguerrePolynomial(n-1, 1), -3): n in [0..25]]; // G. C. Greubel, Feb 24 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Mar 07 2015
STATUS
approved