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A157134 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^n.

%I A157134 #7 Aug 21 2019 21:59:32
%S A157134 1,1,1,1,2,4,7,11,18,33,63,117,211,383,713,1348,2547,4793,9039,17165,
%T A157134 32785,62761,120243,230768,444119,857015,1656931,3207990,6219994,
%U A157134 12079544,23496417,45767352,89256038,174269488,340646238,666604642
%N A157134 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^n.
%F A157134 G.f. satisfies: A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = g.f. of A157135,
%F A157134 where A157135(n) = [x^n] A(x)^(n+1)/(n+1) for n>=0,
%F A157134 and a(n) = [x^n] -1/B(x)^(n-1)/(n-1) for n>1.
%F A157134 From _Paul D. Hanna_, Apr 25 2010: (Start)
%F A157134 G.f. A(x) satisfies the continued fraction:
%F A157134 A(x) = 1/(1- x*A(x)/(1- (x^3-x)*A(x)/(1- x^5*A(x)/(1- (x^7-x^3)*A(x)/(1- x^9*A(x)/(1- (x^11-x^5)*A(x)/(1- x^13*A(x)/(1- (x^15-x^7)*A(x)/(1- ...)))))))))
%F A157134 due to an identity of a partial elliptic theta function.
%F A157134 (End)
%F A157134 From _Paul D. Hanna_, May 05 2010: (Start)
%F A157134 Let A = g.f. A(x) at x=q, then A satisfies the q-series:
%F A157134 A = Sum_{n>=0} q^n*A^n*Product_{k=1..n} (1-q^(4k-3)*A)/(1-q^(4k-1)*A).
%F A157134 (End)
%e A157134 G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 11*x^7 +...
%e A157134 A(x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 +...
%e A157134 A(x)^3 = 1 + 3*x + 6*x^2 + 10*x^3 + 18*x^4 + 36*x^5 + 73*x^6 +...
%e A157134 A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 39*x^4 + 80*x^5 + 168*x^6 +...
%e A157134 where
%e A157134 A(x) = 1 + x*A(x) + x^4*A(x)^2 + x^9*A(x)^3 + x^16*A(x)^4 +...
%o A157134 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,(A=sum(m=0,sqrtint(n),x^(m^2)*A^m)));polcoeff(A,n)}
%Y A157134 Cf. A157135, A157133, A157136.
%Y A157134 Cf. A107595. [From _Paul D. Hanna_, Apr 25 2010]
%K A157134 nonn
%O A157134 0,5
%A A157134 _Paul D. Hanna_, Feb 24 2009
%E A157134 Typo in data corrected by _D. S. McNeil_, Aug 17 2010