A244526 Expansion of f(-x^3, -x^5)^2 in powers of x where f() is Ramanujan's two-variable theta function.

1, 0, 0, -2, 0, -2, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, -2, 2, -2, 0, -2, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 2, -2, 0, 0, 3, 0, 2, -2, 0, 0, 2, 0, 2, 0, 0, -2, 0, 0, 0, -2, 0, -2, 0, 0, 0, -2, 0, 0, 2, 0, 0, -2, 0, -2, 1, 0, 2, 0, 0, -2, 2, -2, 2, 0, 0, 0, 3, 0, 0

Offset: 0

Author

Michael Somos, Jun 29 2014

Keywords

• sign

Examples

			G.f. = 1 - 2*x^3 - 2*x^5 + x^6 + 2*x^8 + x^10 + 2*x^14 - 2*x^17 + 2*x^18 + ...
G.f. = q - 2*q^25 - 2*q^41 + q^49 + 2*q^65 + q^81 + 2*q^113 - 2*q^137 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A244465.

Programs

Mathematica

 QP := QPochhammer; A244526[n_]:= SeriesCoefficient[(QP[q^3, q^8]*QP[q^5, q^8]*QP[q^8])^2, {q, 0, n}]; Table[A244526[n], {n, 0, 50}] (* G. C. Greubel, Dec 25 2017 *)

PARI

{a(n) = (-1)^n * sum(k=0, n, issquare(16*k + 1) * issquare(16*(n-k) + 1))};

Formula

Euler transform of period 8 sequence [ 0, 0, -2, 0, -2, 0, 0, -2, ...].

G.f.: f(-x^3, -x^5)^2 = (Sum_{k in Z} (-1)^k * x^(4*k^2 - k))^2.

Convolution square of A244465.

a(9*n + 4) = a(9*n + 7) = 0. a(49*n + 6) = a(n).