A218102 - OEIS

%I #13 Nov 16 2012 21:25:46

%S 1,1,5,63,1319,40559,1740041,102534291,8332829935,944126513627,

%T 150312711346533,33728725902552987,10685563629182909359,

%U 4790986916169721578103,3046269113896919000444225,2750105392841508250133575939,3528869100728541732126472203599

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

%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 + 5*x^2 + 63*x^3 + 1319*x^4 + 40559*x^5 +...

%e where

%e A(x) = 1 + 2^0*1^1*x*A(x)*exp(-2*1*x*A(x)) + 3^1*2^2*x^2*A(2*x)^2*exp(-3*2*x*A(2*x))/2! + 4^2*3^3*x^3*A(3*x)^3*exp(-4*3*x*A(3*x))/3! + 5^3*4^4*x^4*A(4*x)^4*exp(-5*4*x*A(4*x))/4! + 6^4*5^5*x^5*A(5*x)^5*exp(-6*5*x*A(5*x))/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,x,k*x)^k/k!*exp(-(k+1)*k*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}

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

%Y Cf. A219220, A217900.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 16 2012