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%I #6 Mar 14 2013 13:03:52
%S 1,1,2,13,132,2492,76726,4048401,360486616,54950141846,14338767268684,
%T 6424397920197266,4947731418324541980,6554636080888858780850,
%U 14947781374271898418583534,58699996835841575449007944393,397110307362512858324163841229032
%N O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A((2*n-1)*x)^n/n! * exp(-n*x*A((2*n-1)*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)).
%F a(n) == 1 (mod 2) when n = 2^m-1 for m>=0, and a(n) == 0 (mod 2) otherwise.
%e O.g.f.: A(x) = 1 + x + 2*x^2 + 13*x^3 + 132*x^4 + 2492*x^5 + 76726*x^6 +...
%e where
%e A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(3*x)^2/2!*exp(-2*x*A(3*x)) + 3^3*x^3*A(5*x)^3/3!*exp(-3*x*A(5*x)) + 4^4*x^4*A(7*x)^4/4!*exp(-4*x*A(7*x)) + 5^5*x^5*A(9*x)^5/5!*exp(-5*x*A(9*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^k*x^k*subst(A, x, (2*k-1)*x)^k/k!*exp(-k*x*subst(A, x, (2*k-1)*x)+x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A223076, A218672, A217900.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 14 2013