A229894 Expansion of q^2 * eta(q) / eta(q^49) in powers of q.
1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0
Offset: 0
Keywords
Examples
G.f. = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 - x^35 + ... G.f. = q^-2 - q^-1 - 1 + q^3 + q^5 - q^10 - q^13 + q^20 + q^24 - q^33 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ q] / QPochhammer[ q^49], {q, 0, n}]; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^49 + A), n))};
Formula
Euler transform of period 49 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, ...].
Given g.f. A(x), then B(q) = q^-2*A(q) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u * v * w * (v^2 - 7) - (w - v) * (v - u) * (u*w - v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 7 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A213598.
G.f.: Product_{k>0} (1 - x^k) / (1 - x^(49*k)).
Convolution inverse of A213598.
a(7*n + 3) = a(7*n + 4) = A(7*n + 6) = 0. a(7*n + 2) = 0 unless n=0.
a(7*n) = A108483(n).
a(n) = -(1/n)*Sum_{k=1..n} A287926(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Jun 16 2017