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A262950
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Expansion of q * chi(-q^3) * chi(-q^13) / (chi(-q) * chi(-q^39)) in powers of q where chi() is a Ramanujan theta function.
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2
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1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 6, 7, 8, 9, 11, 12, 14, 15, 17, 20, 22, 25, 28, 31, 34, 38, 43, 48, 53, 58, 64, 71, 79, 88, 96, 105, 115, 127, 140, 154, 168, 183, 200, 219, 240, 262, 285, 310, 337, 367, 400, 435, 472, 511, 554, 601, 652, 707, 764, 825
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OFFSET
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1,6
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COMMENTS
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LINKS
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FORMULA
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Expansion of eta(q^2) * eta(q^3) * eta(q^13) * eta(q^78) / (eta(q) * eta(q^6) * eta(q^26) * eta(q^39)) in powers of q.
Euler transform of a period 78 sequence.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (v - w^2) - 2 * u*w * (1 + v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (78 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128519.
G.f.: x * Product_{k>0} P(x^k) where P(x) is the 78th cyclotomic polynomial of degree 24.
a(n) ~ exp(2*Pi*sqrt(2*n/39)) / (2^(3/4) * 39^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 29 2019
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EXAMPLE
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G.f. = q + q^2 + q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + 3*q^8 + 3*q^9 + 3*q^10 + ...
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MATHEMATICA
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eta[q_] := q^(1/24)*QPochhammer[q]; b := eta[q^2]*eta[q^3]*eta[q^13]* eta[q^78]/(eta[q]*eta[q^6]*eta[q^26]*eta[q^39]); a:= CoefficientList[ Series[q*b, {q, 0, 80}], q]; Drop[Table[a[[n]], {n, 1, 80}], 2] (* G. C. Greubel, Jul 03 2018 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(39*k)) / ((1 + x^(3*k)) * (1 + x^(13*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2019 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^13 + A) * eta(x^78 + A) / (eta(x + A) * eta(x^6 + A) * eta(x^26 + A) * eta(x^39 + 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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