%I #31 Sep 29 2019 07:21:29
%S 1,1,2,9,63,659,9833,206961,6133990,256650268,15213478000,
%T 1281205909177,153588353066135,26245044813624300,6399076697684238375,
%U 2227912079081482302977,1108302173165578509079527,788171767077184315422131588,801638519723021288783092512047
%N O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x)^n/n! * exp(-n*x*A(n*x)).
%C Compare to the LambertW identities:
%C (1) Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).
%C (2) Sum_{n>=0} n^n * x^n * C(x)^n/n! * exp(-n*x*C(x)) = C(x), where C(x) = 1 + x*C(x)^2 is the o.g.f. of the Catalan numbers (A000108).
%H Paul D. Hanna, <a href="/A218672/b218672.txt">Table of n, a(n) for n = 0..50</a>
%e O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 659*x^5 + 9833*x^6 +...
%e where
%e A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2*x)^2/2!*exp(-2*x*A(2*x)) + 3^3*x^3*A(3*x)^3/3!*exp(-3*x*A(3*x)) + 4^4*x^4*A(4*x)^4/4!*exp(-4*x*A(4*x)) + 5^5*x^5*A(5*x)^5/5!*exp(-5*x*A(5*x)) +...
%e simplifies to a power series in x with integer coefficients.
%t a[n_] := Module[{A}, A[x_] = 1 + x; For[i = 1, i <= n, i++, A[x_] = Sum[If[k == 0, 1, k^k] x^k A[k x]^k/k! Exp[-k x A[k x] + x O[x]^i] // Normal, {k, 0, n}]]; Coefficient[ A[x], x, n]];
%t a /@ Range[0, 18] (* _Jean-François Alcover_, Sep 29 2019 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A,x,k*x)^k/k!*exp(-k*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A218673, A218674, A218675, A218676, A218681.
%Y Cf. A217900, A218670, A218667, A218668, A218669, A134055.
%Y Cf. A193363, A221409, A221410, A221411, A221412, A221413.
%Y Cf. A209276, A209277.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 04 2012