A303668 - OEIS

%I #8 Apr 30 2018 08:43:36

%S 1,2,3,5,8,12,19,30,46,71,111,172,266,413,640,991,1537,2383,3692,5722,

%T 8869,13745,21303,33018,51172,79308,122917,190503,295251,457597,

%U 709207,1099165,1703546,2640245,4091988,6341979,9829132,15233702,23609994,36592010,56712212,87895562

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

%C Partial sums of A023361.

%H Alois P. Heinz, <a href="/A303668/b303668.txt">Table of n, a(n) for n = 0..5254</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 - x)*(1 - Sum_{k>=1} x^(k*(k+1)/2))).

%p b:= proc(n) option remember; `if`(n=0, 1,

%p       add(`if`(issqr(8*j+1), b(n-j), 0), j=1..n))

%p     end:

%p a:= proc(n) option remember;

%p       `if`(n<0, 0, b(n)+a(n-1))

%p     end:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 28 2018

%t nmax = 41; CoefficientList[Series[1/((1 - x) (2 - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)))), {x, 0, nmax}], x]

%t nmax = 41; CoefficientList[Series[1/((1 - x) (1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}])), {x, 0, nmax}], x]

%t 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 Cf. A000217, A010054, A023361, A302835, A303667.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 28 2018