A221102 E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} log( A(x)^n )^n / n!.

1, 1, 2, 15, 200, 3760, 91224, 2709812, 95224912, 3863456064, 177719432160, 9139465292232, 519589955270880, 32357519212143216, 2190546095240274576, 160174321738326207720, 12580536897832727328384, 1056318093163793284320000, 94419894211308238918810752

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Formula

E.g.f. satisfies: A(x) = 1 + x/(1 + LambertW(-log(A(x)))).

a(n) ~ n^(n-1) * sqrt(s*(s-1)^3/((r-1)*s+2*s^2-1)) / (exp(n) * r^n), where s = 1.30989890437082404330133094063455155... is the root of the equation 1 + (s-1)*LambertW(-log(s)) / (s*log(s)*(1 + LambertW(-log(s)))^2) = 0, and r = (s-1)*(1+LambertW(-log(s))) = 0.1845269281080403527171896528382422... - Vaclav Kotesovec, Feb 28 2014

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 15*x^3/3! + 200*x^4/4! + 3760*x^5/5! +...
where
A(x) = 1 + x + x*log(A(x)) + 2^2*x*log(A(x))^2/2! + 3^3*x*log(A(x))^3/3! + 4^4*x*log(A(x))^4/4! + 5^5*x*log(A(x))^5/5! +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, log( subst(A^m, x, x+x*O(x^n)) )^m/m!)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A046249 A042355 A208467 * A351920 A319466 A020557

Adjacent sequences: A221099 A221100 A221101 * A221103 A221104 A221105

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2013

Status

approved