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A246713
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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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2
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1, 2, 4, 3, 0, -5, -7, -2, 8, 16, 12, -7, -29, -35, -10, 37, 70, 53, -21, -106, -126, -38, 119, 226, 164, -70, -326, -378, -106, 353, 652, 469, -189, -885, -1015, -290, 910, 1664, 1179, -483, -2205, -2492, -692, 2212, 3998, 2809, -1120, -5119, -5754, -1598
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OFFSET
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-1,2
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COMMENTS
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A generator (Hauptmodul) of the function field associated with the congruence subgroup Gamma_1(7).
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LINKS
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FORMULA
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Euler transform of period 7 sequence [ 2, 1, -3, -3, 1, 2, 0, ...].
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).
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EXAMPLE
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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 + ...
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MATHEMATICA
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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}]]];
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}];
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PROG
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(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))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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