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A157133 G.f. satisfies: A(x) = Sum_{n>=0} x^(n(n+1)/2) * A(x)^n.

%I A157133 #3 Mar 30 2012 18:37:16
%S A157133 1,1,1,2,4,7,14,30,62,129,278,604,1313,2883,6386,14203,31733,71272,
%T A157133 160725,363670,825653,1880351,4293985,9830499,22558939,51880565,
%U A157133 119552907,276012657,638348123,1478749229,3430799333,7971134523
%N A157133 G.f. satisfies: A(x) = Sum_{n>=0} x^(n(n+1)/2) * A(x)^n.
%F A157133 Contribution from _Paul D. Hanna_, Apr 25 2010: (Start)
%F A157133 G.f. A(x) satisfies the continued fraction:
%F A157133 A(x) = 1/(1- x*A(x)/(1- (x^2-x)*A(x)/(1- x^3*A(x)/(1- (x^4-x^2)*A(x)/(1- x^5*A(x)/(1- (x^6-x^3)*A(x)/(1- x^7*A(x)/(1- (x^8-x^4)*A(x)/(1- ...)))))))))
%F A157133 due to an identity of a partial elliptic theta function.
%F A157133 (End)
%e A157133 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 14*x^6 + 30*x^7 +...
%e A157133 A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 13*x^4 + 26*x^5 + 54*x^6 +...
%e A157133 A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 30*x^4 + 66*x^5 + 145*x^6 +...
%e A157133 A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 59*x^4 + 140*x^5 + 326*x^6 +...
%e A157133 where
%e A157133 A(x) = 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 +...
%o A157133 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,(A=sum(m=0,sqrtint(2*n+1),x^(m*(m+1)/2)*A^m)));polcoeff(A,n)}
%Y A157133 Cf. A157134, A157135, A157136.
%Y A157133 Cf. A121690. [From _Paul D. Hanna_, Apr 25 2010]
%K A157133 nonn
%O A157133 0,4
%A A157133 _Paul D. Hanna_, Feb 24 2009