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A303909 Expansion of 2*(1 - x)/(3 - theta_3(x)), where theta_3() is the Jacobi theta function.

%I A303909 #7 Feb 16 2025 08:33:54
%S A303909 1,0,0,0,1,1,1,1,2,4,5,6,8,13,19,26,36,51,74,105,148,208,296,421,597,
%T A303909 846,1198,1699,2409,3417,4843,6865,9732,13799,19566,27739,39325,55749,
%U A303909 79041,112063,158877,225241,319331,452734,641866,910001,1290137,1829079,2593169,3676457,5212266
%N A303909 Expansion of 2*(1 - x)/(3 - theta_3(x)), where theta_3() is the Jacobi theta function.
%C A303909 First differences of A006456.
%H A303909 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A303909 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A303909 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A303909 G.f.: (1 - x)/(1 - Sum_{k>=1} x^(k^2)).
%p A303909 b:= proc(n) option remember; `if`(n<0, 0,
%p A303909       `if`(n=0, 1, add(b(n-j^2), j=1..isqrt(n))))
%p A303909     end:
%p A303909 a:= n-> b(n)-`if`(n=0, 0, b(n-1)):
%p A303909 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 02 2018
%t A303909 nmax = 50; CoefficientList[Series[2 (1 - x)/(3 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
%t A303909 nmax = 50; CoefficientList[Series[(1 - x)/(1 - Sum[x^k^2, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A303909 a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; Differences[Table[a[n], {n, -1, 50}]]
%Y A303909 Cf. A000290, A006456, A078134, A303667.
%K A303909 nonn
%O A303909 0,9
%A A303909 _Ilya Gutkovskiy_, May 02 2018