cp's OEIS Frontend

A229894 Expansion of q^2 * eta(q) / eta(q^49) in powers of q.

%I A229894 #19 Jan 10 2023 05:06:27
%S A229894 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,
%T A229894 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,
%U A229894 -1,0,0,0,0,0,-1,1,0,0,0,1,0,-1,0,0,0,0,0
%N A229894 Expansion of q^2 * eta(q) / eta(q^49) in powers of q.
%H A229894 Seiichi Manyama, <a href="/A229894/b229894.txt">Table of n, a(n) for n = 0..10000</a>
%F A229894 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, ...].
%F A229894 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).
%F A229894 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.
%F A229894 G.f.: Product_{k>0} (1 - x^k) / (1 - x^(49*k)).
%F A229894 Convolution inverse of A213598.
%F A229894 a(7*n + 3) = a(7*n + 4) = A(7*n + 6) = 0. a(7*n + 2) = 0 unless n=0.
%F A229894 a(7*n) = A108483(n).
%F A229894 a(n) = -(1/n)*Sum_{k=1..n} A287926(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Jun 16 2017
%e A229894 G.f. = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 - x^35 + ...
%e A229894 G.f. = q^-2 - q^-1 - 1 + q^3 + q^5 - q^10 - q^13 + q^20 + q^24 - q^33 + ...
%t A229894 a[ n_] := SeriesCoefficient[ QPochhammer[ q] / QPochhammer[ q^49], {q, 0, n}];
%o A229894 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^49 + A), n))};
%Y A229894 Cf. A108483, A213598, A287926.
%K A229894 sign
%O A229894 0,99
%A A229894 _Michael Somos_, Oct 02 2013