OFFSET
0,3
COMMENTS
Radius of convergence of A(x): r = (1/2)*exp(-3/4) = 0.23618..., where A(r) = exp(3/4) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..364
FORMULA
a(n) = n! * [x^n] exp(x+x^2)^(n+1)/(n+1).
a(n) = n! * Sum_{k=floor(n/2)..n} binomial(k,n-k)*(n+1)^(k-1)/k!. - Vladimir Kruchinin, Aug 04 2011
a(n) ~ 2^(n+1/2) * n^(n-1) / (sqrt(3) * exp(n/4 - 3/4)). - Vaclav Kotesovec, Jan 24 2014
E.g.f.: (1/x) * Series_Reversion( x*exp(-x*(1 + x)) ). - _ Seiichi Manyama_, Sep 23 2024
MATHEMATICA
Table[n!*SeriesCoefficient[(E^(x+x^2))^(n+1)/(n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2014 *)
PROG
(PARI) a(n)=n!*polcoeff(exp(x+x^2)^(n+1)+x*O(x^n), n, x)/(n+1)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 07 2003
STATUS
approved