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FORMULA
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GENERATING FUNCTIONS.
(1) A(x) = (1-x) * Sum_{n>=0} x^n * (1 + (1-x)^2)^(n^2).
(2) A(x) = (1-x) * Sum_{n>=0} x^n*q^n * Product_{k=1..n} (1 - q^(4*k-3)*x) / (1 - q^(4*k-1)*x) where q = 1 + (1-x)^2, due to a q-series identity.
(3) A(x) = (1-x)/(1 - q*x/(1 - q*(q^2-1)*x/(1 - q^5*x/(1 - q^3*(q^4-1)*x/(1 - q^9*x/(1- q^5*(q^6-1)*x/(1 - q^13*x/(1 - q^7*(q^8-1)*x/(1 - ...))))))))) where q = 1 + (1-x)^2, a continued fraction due to an identity of a partial elliptic theta function.
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PROG
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(PARI) /* G.f. by Definition: */
{a(n) = my(A = (1-x) * sum(m=0, 2*n, x^m * (1 + (1-x)^2 +x*O(x^n) )^(m^2))); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Continued fraction expression: */
{a(n) = my(CF=1, q = 1 + (1-x)^2 +x*O(x^n)); for(k=0, n, CF = 1/(1 - q^(4*n-4*k+1)*x/(1 - q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*x*CF)) ); polcoeff((1-x)*CF, n, x)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* G.f. by q-series identity: */
{a(n) = my(A =1, q = 1 + (1-x)^2 +x*O(x^n)); A = (1-x) * sum(m=0, 2*n, x^m*q^m * prod(k=1, m, (1 - x*q^(4*k-3)) / (1 - x*q^(4*k-1) +x*O(x^n)) )); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))
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