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A303668 Expansion of 1/((1 - x)*(2 - theta_2(sqrt(x))/(2*x^(1/8)))), where theta_2() is the Jacobi theta function.

%I A303668 #9 Feb 16 2025 08:33:54
%S A303668 1,2,3,5,8,12,19,30,46,71,111,172,266,413,640,991,1537,2383,3692,5722,
%T A303668 8869,13745,21303,33018,51172,79308,122917,190503,295251,457597,
%U A303668 709207,1099165,1703546,2640245,4091988,6341979,9829132,15233702,23609994,36592010,56712212,87895562
%N A303668 Expansion of 1/((1 - x)*(2 - theta_2(sqrt(x))/(2*x^(1/8)))), where theta_2() is the Jacobi theta function.
%C A303668 Partial sums of A023361.
%H A303668 Alois P. Heinz, <a href="/A303668/b303668.txt">Table of n, a(n) for n = 0..5254</a>
%H A303668 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A303668 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A303668 G.f.: 1/((1 - x)*(1 - Sum_{k>=1} x^(k*(k+1)/2))).
%p A303668 b:= proc(n) option remember; `if`(n=0, 1,
%p A303668       add(`if`(issqr(8*j+1), b(n-j), 0), j=1..n))
%p A303668     end:
%p A303668 a:= proc(n) option remember;
%p A303668       `if`(n<0, 0, b(n)+a(n-1))
%p A303668     end:
%p A303668 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 28 2018
%t A303668 nmax = 41; CoefficientList[Series[1/((1 - x) (2 - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)))), {x, 0, nmax}], x]
%t A303668 nmax = 41; CoefficientList[Series[1/((1 - x) (1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}])), {x, 0, nmax}], x]
%t A303668 a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, 8 k + 1] a[n - k], {k, 1, n}]/2; Accumulate[Table[a[n], {n, 0, 41}]]
%Y A303668 Cf. A000217, A010054, A023361, A302835, A303667.
%K A303668 nonn
%O A303668 0,2
%A A303668 _Ilya Gutkovskiy_, Apr 28 2018