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URL: https://oeis.org/A094023

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A094023
Expansion of eta(q^6) * eta(q^10) / (eta(q) * eta(q^15)) in powers of q.
10
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
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 10, 13.
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
Sequence in context: A065094 A145728 A145786 * A123630 A326977 A035967
KEYWORD
nonn
AUTHOR
Michael Somos, Apr 22 2004
STATUS
approved