A219264 - OEIS

%I #4 Nov 16 2012 20:53:04

%S 1,1,8,128,3259,120082,6151625,433404057,42180568185,5720993700540,

%T 1088246094845838,291276119631119408,109983236494820652007,

%U 58741463418913578672779,44466318283501559718838424,47771843216826858235974983400,72930986725295232949801895385998

%N  O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^2*x)^n/n! * exp(-n^2*x*A(n*x)).

%C  Compare to the LambertW identity:

%C Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

%e  O.g.f.: A(x) = 1 + x + 8*x^2 + 128*x^3 + 3259*x^4 + 120082*x^5 +...

%e where

%e A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(2*x)^2/2!*exp(-2^2*x*A(2*x)) + 3^6*x^3*A(3*x)^3/3!*exp(-3^2*x*A(3*x)) + 4^8*x^4*A(4*x)^4/4!*exp(-4^2*x*A(4*x)) + 5^10*x^5*A(5*x)^5/5!*exp(-5^2*x*A(5*x)) +...

%e simplifies to a power series in x with integer coefficients.

%o  (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^(2*k)*x^k*subst(A,x,k*x)^k/k!*exp(-k^2*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}

%o for(n=0,25,print1(a(n),", "))

%Y  Cf. A219265, A218681, A218672, A219184, A217900.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 16 2012