login
G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n^2) / Product_{k=1..n} (1 + k*x*A(x)^n).
1

%I #3 Feb 04 2013 18:12:55

%S 1,1,2,9,54,392,3264,30375,311482,3492134,42613740,564395954,

%T 8094807168,125423821396,2093539627292,37521869868373,719483654260090,

%U 14705046942685816,319171681858506880,7331367124418082012,177646903957002411656,4527740283395695051578

%N G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n^2) / Product_{k=1..n} (1 + k*x*A(x)^n).

%C Compare the g.f. to the identities:

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

%C (2) C(x) = Sum_{n>=0} n!*x^n*C(x)^n / Product_{k=1..n} (1 + k*x*C(x)), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 54*x^4 + 392*x^5 + 3264*x^6 +...

%e where

%e A(x) = 1 + x*A(x)/(1+x*A(x)) + 2!*x^2*A(x)^4/((1+x*A(x)^2)*(1+2*x*A(x)^2)) + 3!*x^3*A(x)^9/((1+x*A(x)^3)*(1+2*x*A(x)^3)*(1+3*x*A(x)^3)) + 4!*x^4*A(x)^16/((1+x*A(x)^4)*(1+2*x*A(x)^4)*(1+3*x*A(x)^4)*(1+4*x*A(x)^4)) +...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*A^(m^2)/prod(k=1, m, 1+k*x*(A+x*O(x^n))^m))); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A222013, A221585.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 04 2013