A138764 E.g.f. A(x) equals the inverse function of log(x)/(x + x^2).

1, 2, 16, 248, 5824, 184512, 7379200, 356956672, 20274442240, 1322971320320, 97542692798464, 8020249539919872, 727662513046159360, 72215332738579824640, 7782298855258810482688, 905031449967822916026368

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

Cf. A138860.

Sequence in context: A188560 A012462 A012457 * A009833 A009044 A019318

Adjacent sequences: A138761 A138762 A138763 * A138765 A138766 A138767

Keyword

nonn

Author

Paul D. Hanna, Mar 29 2008

Status

approved