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A193114 G.f. A(x) satisfies 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(n+1).

%I A193114 #15 Apr 13 2023 08:09:51
%S A193114 1,1,2,5,13,37,111,345,1103,3604,11977,40356,137543,473317,1642258,
%T A193114 5738828,20179338,71346433,253485527,904536366,3240418665,11649734335,
%U A193114 42017535527,151992797355,551298507620,2004602732825,7305747551718,26682235709115
%N A193114 G.f. A(x) satisfies 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(n+1).
%H A193114 Robert Israel, <a href="/A193114/b193114.txt">Table of n, a(n) for n = 0..1600</a>
%F A193114 The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A193114 (1) 1 = Sum_{n>=0} (-x)^n*A(x)^(n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x))/(1 + x^(4*k-1)*A(x));
%F A193114 (2) 1 = A(x)/(1 + x*A(x)/(1 - x*(1-x^2)*A(x)/(1 + x^5*A(x)/(1 - x^3*(1-x^4)*A(x)/(1 + x^9*A(x)/(1 - x^5*(1-x^6)*A(x)/(1 + x^13*A(x)/(1 - x^7*(1-x^8)*A(x)/(1- ...))))))))) (continued fraction);
%F A193114 due to identities of a partial elliptic theta function.
%e A193114 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 111*x^6 + ...
%e A193114 which satisfies:
%e A193114 1 = A(x) - x*A(x)^2 + x^4*A(x)^3 - x^9*A(x)^4 + x^16*A(x)^5 -+ ...
%e A193114 Related expansions.
%e A193114 A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 40*x^4 + 120*x^5 + 373*x^6 + ...
%e A193114 A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 87*x^4 + 276*x^5 + 893*x^6 + ...
%p A193114 e36:= 1 - add((-x)^(n^2)*a^(n+1),n=0..6):
%p A193114 S:= series(RootOf(e36,a),x,37):
%p A193114 seq(coeff(S,x,i),i=0..36); # _Robert Israel_, Apr 10 2023
%o A193114 (PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(#A)+1, (-x)^(m^2)*Ser(A)^(m+1)), #A-1)); if(n<0, 0, A[n+1])}
%Y A193114 Cf. A193111, A193112, A193113, A193115, A193116.
%K A193114 nonn
%O A193114 0,3
%A A193114 _Paul D. Hanna_, Jul 16 2011