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O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(n-1) * (n*x)^n * A(n*x)^(2*n)/n! * exp(-(n+1)*n*x*A(n*x)^2).
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%I #12 Nov 16 2012 21:28:02

%S 1,1,6,91,2306,86576,4570570,333164243,33547502582,4704103190166,

%T 925622587155708,256758944391842662,100693326907873920440,

%U 55964816627849652514434,44167198051129910003931850,49561249392391287991062025027,79164926515567602205248823277126

%N O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(n-1) * (n*x)^n * A(n*x)^(2*n)/n! * exp(-(n+1)*n*x*A(n*x)^2).

%C Compare the g.f. to the LambertW identity:

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

%e O.g.f.: A(x) = 1 + x + 6*x^2 + 91*x^3 + 2306*x^4 + 86576*x^5 +...

%e where

%e A(x) = 1 + 2^0*1^1*x*A(x)^2*exp(-2*1*x*A(x)^2) + 3^1*2^2*x^2*A(2*x)^4*exp(-3*2*x*A(2*x)^2)/2! + 4^2*3^3*x^3*A(3*x)^6*exp(-4*3*x*A(3*x)^2)/3! + 5^3*4^4*x^4*A(4*x)^8*exp(-5*4*x*A(4*x)^2)/4! + 6^4*5^5*x^5*A(5*x)^10*exp(-6*5*x*A(5*x)^2)/5! +...

%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+1)^(k-1)*k^k*x^k*subst(A^2,x,k*x)^k/k!*exp(-(k+1)*k*x*subst(A^2,x,k*x)+x*O(x^n))));polcoeff(A,n)}

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

%Y Cf. A218102, A217900.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 16 2012