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Expansion of 1/(1 - x*(1 + theta_3(x))/2), where theta_3() is the Jacobi theta function.
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%I #7 Feb 16 2025 08:33:53

%S 1,1,2,3,5,9,15,26,44,75,129,220,377,644,1101,1883,3219,5506,9414,

%T 16098,27527,47069,80488,137630,235343,402427,688134,1176685,2012085,

%U 3440591,5883279,10060183,17202533,29415676,50299693,86010564,147074801,251492331,430042340,735356089,1257431006

%N Expansion of 1/(1 - x*(1 + theta_3(x))/2), where theta_3() is the Jacobi theta function.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F G.f.: 1/(1 - x*Sum_{k>=0} x^(k^2)).

%F a(0) = 1; a(n) = Sum_{k=1..n} A010052(k-1)*a(n-k).

%t nmax = 40; CoefficientList[Series[1/(1 - x (1 + EllipticTheta[3, 0, x])/2), {x, 0, nmax}], x]

%t nmax = 40; CoefficientList[Series[1/(1 - x Sum[x^k^2, {k, 0, nmax}]), {x, 0, nmax}], x]

%Y Antidiagonal sums of A045847.

%Y Cf. A010052, A032803, A181649, A302019.

%K nonn,changed

%O 0,3

%A _Ilya Gutkovskiy_, Mar 30 2018