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

%I #19 May 05 2017 12:33:51

%S 1,1,2,3,5,7,10,14,20,27,36,48,63,82,106,137,175,222,280,352,439,546,

%T 676,834,1024,1253,1528,1857,2250,2718,3276,3936,4718,5640,6728,8006,

%U 9507,11266,13324,15726,18526,21786,25574,29970,35064,40961,47774,55638

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

%H Seiichi Manyama, <a href="/A094023/b094023.txt">Table of n, a(n) for n = 0..10000</a>

%F 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.

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

%F 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, ...].

%F 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.

%F Convolution inverse of A131797.

%F a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 08 2015

%e 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 + ...

%t 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 *)

%t 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 *)

%o (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))};

%Y Cf. A058618, A131797.

%K nonn

%O 0,3

%A _Michael Somos_, Apr 22 2004

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