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A123648
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Expansion of eta(q^4) * eta(q^28) / (eta(q) * eta(q^7)) in powers of q.
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3
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1, 1, 2, 3, 4, 6, 9, 13, 17, 24, 32, 42, 56, 73, 96, 123, 158, 201, 254, 320, 402, 504, 624, 774, 955, 1172, 1436, 1755, 2138, 2592, 3140, 3789, 4560, 5478, 6564, 7851, 9362, 11146, 13240, 15696, 18574, 21942, 25880, 30456, 35796, 42000, 49196, 57546
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of q / (chi(-q) * chi(-q^2) * chi(-q^7) * chi(-q^14)) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Nov 11 2013
Euler transform of period 28 sequence [ 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v*(1 + 2*(u+v) + 4*u*v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u*v * (1 + 2*u) * (1 + 2*v) * (1 + u+v + 4*u*v). - Michael Somos, Nov 11 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A161970. - Michael Somos, Nov 11 2013
a(n) ~ exp(2*Pi*sqrt(n/7)) / (8 * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018
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EXAMPLE
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G.f. = q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 6*q^6 + 9*q^7 + 13*q^8 + 17*q^9 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ q^4] QPochhammer[ q^28] / (QPochhammer[ q] QPochhammer[ q^7]), {q, 0, n}]
a[ n_] := SeriesCoefficient[ QPochhammer[-q, q] QPochhammer[-q^2, q^2] QPochhammer[-q^7, q^7] QPochhammer[-q^14, q^14], {q, 0, n}]
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^28 + A) / (eta(x + A) * eta(x^7 + 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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