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A303908 Expansion of 1/(2 + x - theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.

%I A303908 #11 Feb 16 2025 08:33:54
%S A303908 1,0,0,1,0,0,2,0,0,3,1,0,5,2,0,9,5,0,15,10,1,27,20,3,46,40,9,80,78,22,
%T A303908 139,152,51,242,290,114,427,550,247,753,1034,525,1340,1933,1092,2396,
%U A303908 3602,2237,4312,6685,4519,7813,12380,9027,14239,22877,17866,26110,42214,35072,48123,77829,68379
%N A303908 Expansion of 1/(2 + x - theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.
%C A303908 Number of compositions (ordered partitions) of n into triangular numbers > 1.
%H A303908 Seiichi Manyama, <a href="/A303908/b303908.txt">Table of n, a(n) for n = 0..10000</a>
%H A303908 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A303908 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A303908 G.f.: 1/(1 - Sum_{k>=2} x^(k*(k+1)/2)).
%p A303908 a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A303908       add(a(n-j*(j+1)/2), j=2..isqrt(2*n))))
%p A303908     end:
%p A303908 seq(a(n), n=0..80);  # _Alois P. Heinz_, May 02 2018
%t A303908 nmax = 62; CoefficientList[Series[1/(2 + x - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))), {x, 0, nmax}], x]
%t A303908 nmax = 62; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2), {k, 2, nmax}]), {x, 0, nmax}], x]
%t A303908 a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, 8 k + 1] a[n - k], {k, 2, n}]/2; Table[a[n], {n, 0, 62}]
%Y A303908 Cf. A000217, A023361, A212804, A280542, A303668, A303906, A303907.
%K A303908 nonn
%O A303908 0,7
%A A303908 _Ilya Gutkovskiy_, May 02 2018