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G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2*x^n * A(x)^(n^2).
5

%I #6 Jun 16 2014 19:20:19

%S 1,2,6,30,194,1442,11782,103102,951554,9173186,91780614,948985822,

%T 10110931650,110794764642,1247186300934,14412811665278,

%U 170949340705794,2081185257723778,26012832364535814,333929563132811678,4404347475363755714,59705917899701420834,832080588205468939782

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

%H Paul D. Hanna, <a href="/A176719/b176719.txt">Table of n, a(n) for n = 0..100</a>

%F Contribution from _Paul D. Hanna_, May 11 2010: (Start)

%F Given g.f. A(x), then Q = A(-x^2) satisfies:

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

%F due to a q-series expansion for the Jacobi theta_4 function. (End)

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 30*x^3 + 194*x^4 + 1442*x^5 +...

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

%e Contribution from _Paul D. Hanna_, May 11 2010: (Start)

%e Given g.f. A(x), then Q = A(-x^2) satisfies the q-series:

%e Q/(1-x) = 1 + x*(xQ;Q)_1/(-xQ;Q)_1 + x^2*(xQ;Q)_2/(-xQ;Q)_2 +...

%e where the q-Pochhammer symbol (x;q)_n = Product_{k=0..n-1} (1-x*q^k). (End)

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

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A107595, A176720.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 25 2010

%E Edited by _Paul D. Hanna_, Apr 27 2010