A094023 Expansion of eta(q^6) * eta(q^10) / (eta(q) * eta(q^15)) in powers of q.

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 106, 137, 175, 222, 280, 352, 439, 546, 676, 834, 1024, 1253, 1528, 1857, 2250, 2718, 3276, 3936, 4718, 5640, 6728, 8006, 9507, 11266, 13324, 15726, 18526, 21786, 25574, 29970, 35064, 40961, 47774, 55638

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + 2*v^2 - 2*u*v^2.

G.f. A(x) satisfies A(x) + A(-x) = 2*A(x^2)^2, (1 - A(x)) * (1 - A(-x)) = 1 - A(x^2).

Euler transform of period 30 sequence [ 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058618.

Convolution inverse of A131797.

a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 08 2015

Example

G.f. = 1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 10*q^6 + 14*q^7 + 20*q^8 + ...

Mathematica

nmax = 60; CoefficientList[Series[Product[(1-x^(6*k)) * (1-x^(10*k)) / ((1-x^k) * (1-x^(15*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)
QP = QPochhammer; s = QP[q^6]*(QP[q^10]/(QP[q]*QP[q^15])) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) * eta(x^10 + A) / eta(x + A) / eta(x^15 + A), n))};

Crossrefs

Cf. A058618, A131797.

Sequence in context: A065094 A145728 A145786 * A123630 A326977 A035967

Adjacent sequences: A094020 A094021 A094022 * A094024 A094025 A094026

Keyword

nonn

Author

Michael Somos, Apr 22 2004

Status

approved