A221411 - OEIS

%I #7 Jan 15 2013 18:36:28

%S 1,1,4,27,325,5553,140103,4993445,253780210,18321882898,1882677322704,

%T 275715048156637,57570654555092091,17152947168669102772,

%U 7295365645092117304955,4430848642167010373612127,3844378527942068170940925685,4766454891141869269974497298382

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

%C Compare to the LambertW identity:

%C Sum_{n>=0} (n+2)^n * x^n * G(x)^n/n! * exp(-(n+2)*x*G(x)) = 1/(1 - x*G(x)).

%e O.g.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 325*x^4 + 5553*x^5 + 140103*x^6 +...

%e where

%e A(x) = exp(-2*x) + 3*x*A(x)*exp(-3*x*A(x)) + 4^2*x^2*A(2*x)^2/2!*exp(-4*x*A(2*x)) + 5^3*x^3*A(3*x)^3/3!*exp(-5*x*A(3*x)) + 6^4*x^4*A(4*x)^4/4!*exp(-6*x*A(4*x)) + 7^5*x^5*A(5*x)^5/5!*exp(-7*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, 20, print1(a(n), ", "))

%Y Cf. A218672, A193363, A221409, A221410, A221412, A221413.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 15 2013