A088693 E.g.f: A(x) = f(x*A(x)^2), where f(x) = (1+3*x)*exp(x).

1, 4, 71, 2434, 126117, 8804776, 775425427, 82565249670, 10319537275913, 1481520436347628, 240291243489544191, 43458295155840595306, 8672066947756086825325, 1892794863486905965709136, 448582856421716543783775947, 114720816495997657177701763246

Offset: 0

Author

Paul D. Hanna, Oct 07 2003

Keywords

• nonn

Comments

Radius of convergence of A(x): r = (2/27)*exp(-1/3) = 0.053076..., where A(r) = (3/2)*exp(1/6) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.

Crossrefs

Cf. A088690, A088692.

Programs

Mathematica

Table[n!*SeriesCoefficient[((1+3*x)*E^x)^(2*n+1)/(2*n+1),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 24 2014 *)

PARI

a(n)=n!*polcoeff(((1+3*x)*exp(x))^(2*n+1)+x*O(x^n),n,x)/(2*n+1)

Formula

a(n) = n! * [x^n] ((1+3*x)*exp(x))^(2*n+1)/(2*n+1).

a(n) ~ 3^(3*n+2) * n^(n-1) / (sqrt(7) * 2^(n+2) * exp(2*n/3-1/6)). - Vaclav Kotesovec, Jan 24 2014