A218674 - OEIS

%I #11 Nov 05 2012 12:01:01

%S 1,1,4,34,455,8710,230077,8285224,407456797,27587687551,2596034329278,

%T 342275007167359,63606742005546232,16730509857101195808,

%U 6246818082857455197662,3317816101992338134691233,2510420393373091580780786808,2709148467943025007607468405672

%N O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(3*n)/n! * exp(-n*x*A(n*x)^3).

%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 + 4*x^2 + 34*x^3 + 455*x^4 + 8710*x^5 + 230077*x^6 +...

%e where

%e A(x) = 1 + x*A(x)^3*exp(-x*A(x)^3) + 2^2*x^2*A(2*x)^6/2!*exp(-2*x*A(2*x)^3) + 3^3*x^3*A(3*x)^9/3!*exp(-3*x*A(3*x)^3) + 4^4*x^4*A(4*x)^12/4!*exp(-4*x*A(4*x)^3) + 5^5*x^5*A(5*x)^15/5!*exp(-5*x*A(5*x)^3) +...

%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^k*x^k*subst(A^3,x,k*x)^k/k!*exp(-k*x*subst(A^3,x,k*x)+x*O(x^n))));polcoeff(A,n)}

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

%Y Cf. A218672, A218673, A218675, A218676.

%Y Cf. A217900, A218670, A218667, A218668, A218669, A134055.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 04 2012