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Expansion of f(-x^1, -x^7)^2 in powers of x where f() is Ramanujan's two-variable theta function.
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%I #12 Jul 26 2024 05:02:08

%S 1,-2,1,0,0,0,0,-2,2,0,2,-2,0,0,1,0,0,-2,0,0,1,0,2,-2,0,0,0,-2,2,-2,0,

%T 0,2,0,2,0,0,-2,0,0,0,0,0,0,1,-2,2,0,0,-2,0,0,4,-2,1,-2,0,0,0,-2,0,0,

%U 2,0,0,0,0,-2,0,0,0,0,2,0,2,0,2,-2,0,-2,0

%N Expansion of f(-x^1, -x^7)^2 in powers of x where f() is Ramanujan's two-variable theta function.

%H G. C. Greubel, <a href="/A244560/b244560.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 G.f.: f(-x, -x^7)^2 = (Sum_{k in Z} (-1)^k * x^(4*k^2 - 3*k))^2.

%F Convolution square of A244525.

%F a(9*n) = A244526(n). a(9*n + 3) = a(9*n + 6) = 0. a(49*n + 5) = a(n-1).

%e G.f. = 1 - 2*x + x^2 - 2*x^7 + 2*x^8 + 2*x^10 - 2*x^11 + x^14 - 2*x^17 + ...

%e G.f. = q^9 - 2*q^17 + q^25 - 2*q^65 + 2*q^73 + 2*q^89 - 2*q^97 + q^121 + ...

%t A244560[n_] := SeriesCoefficient[(QPochhammer[q^1, q^8]* QPochhammer[q^7, q^8]*QPochhammer[q^8, q^8])^2, {q, 0, n}]; Table[A244560[n], {n,0,50}] (* _G. C. Greubel_, Jun 17 2017 *)

%o (PARI) {a(n) = (-1)^n * sum(k=0, n, issquare(16*k + 9) * issquare(16*(n-k) + 9))};

%Y Cf. A244525, A244526.

%K sign

%O 0,2

%A _Michael Somos_, Jun 30 2014