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

0,2

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

Formula

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

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}]]

Crossrefs

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

Sequence in context: A240523 A023436 A024567 * A060961 A225393 A243850

Adjacent sequences: A303665 A303666 A303667 * A303669 A303670 A303671

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 28 2018

Status

approved