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A132302
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Expansion of f(-x, -x^5) * f(-x^6) / f(-x)^2 in powers of x where f(, ) and f() are Ramanujan theta functions.
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8
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1, 1, 3, 5, 10, 15, 26, 39, 63, 92, 140, 201, 295, 415, 591, 818, 1140, 1554, 2126, 2861, 3855, 5126, 6816, 8970, 11793, 15372, 20007, 25857, 33356, 42771, 54734, 69683, 88530, 111968, 141312, 177642, 222842, 278557, 347484, 432095, 536230, 663549, 819504
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-1/2) * eta(q^6)^3 / (eta(q) * eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^2) * q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v)^3 - 4 * v^4 * (v - 3*u^2) * (2*v - 3*u^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132301.
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4)*3^(3/2)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
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EXAMPLE
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G.f. = 1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 15*x^5 + 26*x^6 + 39*x^7 + ...
G.f. = q + q^3 + 3*q^5 + 5*q^7 + 10*q^9 + 15*q^11 + 26*q^13 + 39*q^15 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6]^2 / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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