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G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^(n^2)).
1

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

%S 1,1,2,8,38,201,1164,7188,46576,313823,2185642,15668473,115281167,

%T 868757478,6696711294,52757324970,424590429862,3490344692094,

%U 29310836035090,251525003170386,2206548594680093,19798923287905907,181797157100106619

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

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

%e G.f: A(x) = 1 + x + 2*x^2 + 8*x^3 + 38*x^4 + 201*x^5 + 1164*x^6 +...

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

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

%e as well as the logarithmic series:

%e log(A) = x*A + x^2*(-A^2 + 2*A^4)/2 + x^3*(A^3 + 3*A^9)/3 + x^4*(-A^4 - 2*A^8 + 4*A^16)/4 + x^5*(A^5 + 5*A^25)/5 + x^6*(-A^6 + 2*A^12 - 3*A^18 + 6*A^36)/6 +...

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

%Y Cf. A192768.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 09 2011