OFFSET
0,2
COMMENTS
Let r = radius of convergence of A(x), then:
r = 0.116689393840305520533609707610483991781804638898970699779...
A(r) = 1.835037067429188745641951736620284283425600418229813004773...
where A(r) and r satisfy:
A(r) = exp( (1 + A(r))/(1 + 2*A(r)) ) and r = 1/(A(r)*(1 + 2*A(r))).
Also, r = (1-t)^2/(t*(1+t)) where (t/(1-t))^(1+t) = e. [Paul D. Hanna, Oct 03 2020, derived from the asymptotic formula given by Vaclav Kotesovec].
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..331
FORMULA
E.g.f. satisfies: A(x) = exp( x*[A(x) + A(x)^2] ).
a(n) = Sum_{k=0..n} binomial(n,k)*(n+k+1)^(n-1) - Vladeta Jovovic, Mar 31 2008.
a(n) = A138860(n)*2^n.
E.g.f. satisfies: A( x/( exp(x) + exp(2*x) ) ) = exp(x).
a(n) ~ n^(n-1)*(1+r)^n*r^(n+1)/(sqrt(1+3*r)*(1-r)^(2*n+1)*exp(n)), where r = 0.6472709258412625... is the root of the equation (r/(1-r))^(1+r) = e. - Vaclav Kotesovec, Jun 15 2013
EXAMPLE
E.g.f. A(x) = 1 + 2x + 16x^2/2! + 248x^3/3! + 5824x^4/4! +...
MATHEMATICA
Table[Sum[Binomial[n, k]*(n+k+1)^(n-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 15 2013 *)
PROG
(PARI) {a(n) = local(A=1+x); for(i=0, n, A=exp(x*(A+A^2 +x*O(x^n) ))); n! * polcoeff(A, n) }
(PARI) {a(n) = sum(k=0, n, binomial(n, k)*(n+k+1)^(n-1))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 29 2008
STATUS
approved