A244526 - OEIS

%I #9 Mar 12 2021 22:24:47

%S 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,

%T 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,

%U 0,-2,0,-2,1,0,2,0,0,-2,2,-2,2,0,0,0,3,0,0

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

%H G. C. Greubel, <a href="/A244526/b244526.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

%F Convolution square of A244465.

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

%e 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 + ...

%e G.f. = q - 2*q^25 - 2*q^41 + q^49 + 2*q^65 + q^81 + 2*q^113 - 2*q^137 + ...

%t  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 *)

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

%Y Cf. A244465.

%K sign

%O 0,4

%A _Michael Somos_, Jun 29 2014