A192783 - OEIS

%I #5 Mar 30 2012 18:37:27

%S 1,1,3,16,119,1145,13301,180464,2821941,50400230,1022250876,

%T 23424407915,602724515761,17299947151776,550101222059396,

%U 19246320555772626,736247255316380311,30620337253882961105,1377609185722013042566,66750666290443384609574

%N G.f. satisfies: A(x) = Product_{n>=1} 1/(1 - x^n*A(x)^(n^3)).

%F G.f. satisfies: A(x) = exp( Sum_{n>=1} (x^n/n)*Sum_{d|n} d*A(x)^(n*d^2) ).

%e G.f: A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1145*x^5 + 13301*x^6 +...

%e The g.f. A = A(x) satisfies:

%e A = 1/((1 - x*A)*(1 - x^2*A^8)*(1 - x^3*A^27)*(1 - x^4*A^64)*...),

%e as well as the logarithmic series:

%e log(A) = x*A + x^2*(A^2 + 2*A^8)/2 + x^3*(A^3 + 3*A^27)/3 + x^4*(A^4 + 2*A^16 + 4*A^64)/4 + x^5*(A^5 + 5*A^125)/5 + x^6*(A^6 + 2*A^24 + 3*A^54 + 6*A^216)/6  +...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, 1-x^k*(A+x*O(x^n))^(k^3))); polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x^m/m)*sumdiv(m, d, d*(A+x*O(x^n))^(m*d^2))))); polcoeff(A, n)}

%Y Cf. A109085, A192768.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 09 2011