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%I #8 Apr 10 2019 02:07:06
%S 1,3,5,7,11,21,49,133,408,1376,5020,19564,80741,350551,1593066,
%T 7547792,37163568,189662934,1001046684,5453972462,30622950955,
%U 176942133603,1050773432990,6405898358012,40048848677954,256521565555908,1681897617101795,11278819380424173,77301464920178158,541084956406886214,3865540113371340736,28167799470180443028,209238063076396838375,1583562040116769584091,12204247180832799551059,95731651337427271893873
%N G.f. A(x) satisfies: x = Sum_{n>=1} x^n * (1+x)^(n^2/2) / A(x)^(n/2).
%H Paul D. Hanna, <a href="/A325155/b325155.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies the following identities.
%F (1) 1 + x = Sum_{n>=0} x^n * (1+x)^(n^2/2) / A(x)^(n/2).
%F (2) 1 + x = 1/(1 - q*x/(sqrt(A(x)) - q*(q^2-1)*x/(1 - q^5*x/(sqrt(A(x)) - q^3*(q^4-1)*x/(1 - q^9*x/(sqrt(A(x)) - q^5*(q^6-1)*x/(1 - q^13*x/(sqrt(A(x)) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = sqrt(1+x), a continued fraction due to a partial elliptic theta function identity.
%F (3) 1 + x = Sum_{n>=0} x^n * (1+x)^(n/2) / A(x)^(n/2) * Product_{k=1..n} (sqrt(A(x)) - x*sqrt(1+x)^(4*k-3)) / (sqrt(A(x)) - x*sqrt(1+x)^(4*k-1)), due to a q-series identity.
%F (4) A(x) = (1+x)*G(x)^2 where G(x) is the g.f. of A318644.
%e G.f.: A(x) = 1 + 3*x + 5*x^2 + 7*x^3 + 11*x^4 + 21*x^5 + 49*x^6 + 133*x^7 + 408*x^8 + 1376*x^9 + 5020*x^10 + 19564*x^11 + 80741*x^12 + ...
%e such that
%e A(x) = 1 + x*(1+x)^(1/2)/A(x)^(1/2) + x^2*(1+x)^2/A(x) + x^3*(1+x)^(9/2)/A(x)^(3/2) + x^4*(1+x)^8/A(x)^2 + x^5*(1+x)^(25/2)/A(x)^(5/2) + x^6*(1+x)^18/A(x)^3 + x^7*(1+x)^(49/2)/A(x)^(7/2) + x^8*(1+x)^32/A(x)^4 + ...
%e Note that
%e sqrt(A(x))*sqrt(1+x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + ... + A318644(n)*x^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = 2*polcoeff( sum(n=0,#A+1, x^n*(1+x +x*O(x^#A))^(n^2/2) / Ser(A)^(n/2) ),#A)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A318644, A303058.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 06 2019
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