OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..375
FORMULA
a(n) ~ n^(n+1/12)*3^n*exp(-n+2*sqrt(n)-1/2)/sqrt(2) * (1 + 65/(144*sqrt(n))). - Vaclav Kotesovec, Jun 22 2013
From Peter Bala, Jun 14 2014: (Start)
E.g.f.: 1/(1 - 3*x)^(2/3)*exp(3*x/(1 - 3*x)) = 1 + 5*x + 49*x^2/2! + 683*x^3/3! + ....
Dobinski-type formula: a(n) = (3^n/exp(1))*Sum {k >= 0} (n!/k!)* binomial(n + k - 1/3,k - 1/3).
Recurrence equation: a(n) = (6*n - 1)a(n-1) - (n - 1)*(9*n - 12)*a(n-2) with a(0) = 1 and a(1) = 5. (End)
MAPLE
MATHEMATICA
Table[3^n*n!*LaguerreL[n, -1/3, -1], {n, 0, 20}] (* Vaclav Kotesovec, Jun 22 2013 *)
PROG
(PARI) x='x+O('x^30); Vec(serlaplace(1/(1 - 3*x)^(2/3)*exp(3*x/(1 - 3*x)))) \\ G. C. Greubel, May 13 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients( R!(1/(1 - 3*x)^(2/3)*Exp(3*x/(1 - 3*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 13 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Nov 14 2003
EXTENSIONS
Terms a(15) onward added by G. C. Greubel, May 13 2018
STATUS
approved