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A303667 Expansion of 2/((1 - x)*(3 - theta_3(x))), where theta_3() is the Jacobi theta function.

%I A303667 #12 Feb 16 2025 08:33:54
%S A303667 1,2,3,4,6,9,13,18,25,36,52,74,104,147,209,297,421,596,845,1199,1701,
%T A303667 2411,3417,4844,6868,9738,13806,19573,27749,39342,55778,79079,112112,
%U A303667 158944,225342,319479,452941,642152,910404,1290719,1829911,2594344,3678108,5214606,7392970,10481335
%N A303667 Expansion of 2/((1 - x)*(3 - theta_3(x))), where theta_3() is the Jacobi theta function.
%C A303667 Partial sums of A006456.
%H A303667 Alois P. Heinz, <a href="/A303667/b303667.txt">Table of n, a(n) for n = 0..5000</a>
%H A303667 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A303667 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A303667 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A303667 G.f.: 1/((1 - x)*(1 - Sum_{k>=1} x^(k^2))).
%p A303667 b:= proc(n) option remember;
%p A303667       `if`(n=0, 1, add(b(n-i^2), i=1..isqrt(n)))
%p A303667     end:
%p A303667 a:= proc(n) option remember;
%p A303667       `if`(n<0, 0, b(n)+a(n-1))
%p A303667     end:
%p A303667 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 28 2018
%t A303667 nmax = 45; CoefficientList[Series[2/((1 - x) (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x]
%t A303667 nmax = 45; CoefficientList[Series[1/((1 - x) (1 - Sum[x^k^2, {k, 1, nmax}])), {x, 0, nmax}], x]
%t A303667 a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 45}]]
%Y A303667 Cf. A000290, A006456, A010052, A302833, A303668.
%K A303667 nonn
%O A303667 0,2
%A A303667 _Ilya Gutkovskiy_, Apr 28 2018