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

1, 2, 3, 5, 8, 12, 19, 30, 46, 71, 111, 172, 266, 413, 640, 991, 1537, 2383, 3692, 5722, 8869, 13745, 21303, 33018, 51172, 79308, 122917, 190503, 295251, 457597, 709207, 1099165, 1703546, 2640245, 4091988, 6341979, 9829132, 15233702, 23609994, 36592010, 56712212, 87895562

Offset: 0

Author

Ilya Gutkovskiy, Apr 28 2018

Keywords

• nonn

Comments

Partial sums of A023361.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5254
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions
• Index entries for sequences related to compositions

Crossrefs

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

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(`if`(issqr(8*j+1), b(n-j), 0), j=1..n))
    end:
a:= proc(n) option remember;
      `if`(n<0, 0, b(n)+a(n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 28 2018

Mathematica

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

Formula

G.f.: 1/((1 - x)*(1 - Sum_{k>=1} x^(k*(k+1)/2))).