|
%I #10 Jun 19 2023 07:06:12
%S 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 139,152,51,242,290,114,427,550,247,753,1034,525,1340,1933,1092,2396,
%U 3602,2237,4312,6685,4519,7813,12380,9027,14239,22877,17866,26110,42214,35072,48123,77829,68379
%N Expansion of 1/(2 + x - theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.
%C Number of compositions (ordered partitions) of n into triangular numbers > 1.
%H Seiichi Manyama, <a href="/A303908/b303908.txt">Table of n, a(n) for n = 0..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: 1/(1 - Sum_{k>=2} x^(k*(k+1)/2)).
%p a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p add(a(n-j*(j+1)/2), j=2..isqrt(2*n))))
%p end:
%p seq(a(n), n=0..80); # _Alois P. Heinz_, May 02 2018
%t nmax = 62; CoefficientList[Series[1/(2 + x - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))), {x, 0, nmax}], x]
%t nmax = 62; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2), {k, 2, nmax}]), {x, 0, nmax}], x]
%t 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 Cf. A000217, A023361, A212804, A280542, A303668, A303906, A303907.
%K nonn
%O 0,7
%A _Ilya Gutkovskiy_, May 02 2018
|
