OFFSET
0,2
COMMENTS
Radius of convergence of A(x): r = (3^2/4^4)*exp(-1/4) = 0.0273797..., where A(r) = (4/3)*exp(1/12) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.
FORMULA
a(n) = n! * [x^n] ((1+4*x)*exp(x))^(3*n+1)/(3*n+1).
a(n) ~ 16^(2*n+1) * n^(n-1) / (sqrt(13) * 9^(n+1) * exp(3*n/4 - 1/12)). - Vaclav Kotesovec, Jan 24 2014
MATHEMATICA
Table[n!*SeriesCoefficient[((1+4*x)*E^x)^(3*n+1)/(3*n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2014 *)
PROG
(PARI) a(n)=n!*polcoeff(((1+4*x)*exp(x))^(3*n+1)+x*O(x^n), n, x)/(3*n+1)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 07 2003
STATUS
approved