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G.f. A(x) satisfies 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(n+1).
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%I #15 Apr 13 2023 08:09:51

%S 1,1,2,5,13,37,111,345,1103,3604,11977,40356,137543,473317,1642258,

%T 5738828,20179338,71346433,253485527,904536366,3240418665,11649734335,

%U 42017535527,151992797355,551298507620,2004602732825,7305747551718,26682235709115

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

%H Robert Israel, <a href="/A193114/b193114.txt">Table of n, a(n) for n = 0..1600</a>

%F The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (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 (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 due to identities of a partial elliptic theta function.

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 111*x^6 + ...

%e which satisfies:

%e 1 = A(x) - x*A(x)^2 + x^4*A(x)^3 - x^9*A(x)^4 + x^16*A(x)^5 -+ ...

%e Related expansions.

%e A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 40*x^4 + 120*x^5 + 373*x^6 + ...

%e A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 87*x^4 + 276*x^5 + 893*x^6 + ...

%p e36:= 1 - add((-x)^(n^2)*a^(n+1),n=0..6):

%p S:= series(RootOf(e36,a),x,37):

%p seq(coeff(S,x,i),i=0..36); # _Robert Israel_, Apr 10 2023

%o (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 Cf. A193111, A193112, A193113, A193115, A193116.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 16 2011