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Expansion of q^(-1) * f(-q^3, -q^4)^3 / (f(-q^1, -q^6)^2 * f(-q^2, -q^5)) in powers of q where f() is Ramanujan's two-variable theta function.
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%I #16 Feb 16 2025 08:33:23

%S 1,2,4,3,0,-5,-7,-2,8,16,12,-7,-29,-35,-10,37,70,53,-21,-106,-126,-38,

%T 119,226,164,-70,-326,-378,-106,353,652,469,-189,-885,-1015,-290,910,

%U 1664,1179,-483,-2205,-2492,-692,2212,3998,2809,-1120,-5119,-5754,-1598

%N Expansion of q^(-1) * f(-q^3, -q^4)^3 / (f(-q^1, -q^6)^2 * f(-q^2, -q^5)) in powers of q where f() is Ramanujan's two-variable theta function.

%C A generator (Hauptmodul) of the function field associated with the congruence subgroup Gamma_1(7).

%H G. C. Greubel, <a href="/A246713/b246713.txt">Table of n, a(n) for n = -1..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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Euler transform of period 7 sequence [ 2, 1, -3, -3, 1, 2, 0, ...].

%F G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 + v) * (u*v^2 + v^2 - 1) - 2*v * (u + 1) * (v^2 + 2*u*v + v + 3*u).

%F a(n) = A108481(n) unless n=0.

%e G.f. = 1/q + 2 + 4*q + 3*q^2 - 5*q^4 - 7*q^5 - 2*q^6 + 8*q^7 + 16*q^8 + ...

%t a[ n_] := If[ n < -1, 0, With[{m = n + 1}, SeriesCoefficient[ 1/q Product[ (1 - q^k)^{-2, -1, 3, 3, -1, -2, 0}[[Mod[k, 7, 1]]], {k, m}], {q, 0, n}]]];

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^3, q^7] QPochhammer[ q^4, q^7])^3 / (QPochhammer[ q^1, q^7]^2 QPochhammer[ q^2, q^7] QPochhammer[ q^5, q^7] QPochhammer[ q^6, q^7]^2), {q, 0, n}];

%o (PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[ 0, -2, -1, 3, 3, -1, -2][k%7 + 1]), n))};

%Y Cf. A108481.

%K sign,changed

%O -1,2

%A _Michael Somos_, Sep 02 2014