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A121593 Expansion of (eta(q^7) / eta(q))^4 in powers of q.

%I A121593 #26 Apr 02 2019 00:39:03
%S A121593 1,4,14,40,105,252,574,1236,2564,5124,9948,18788,34685,62664,111132,
%T A121593 193672,332325,561996,937958,1546132,2519825,4062888,6486008,10257324,
%U A121593 16079389,24996636,38555216,59025820,89728900,135486960,203274344
%N A121593 Expansion of (eta(q^7) / eta(q))^4 in powers of q.
%C A121593 G.f. A(q) is denoted by tau(q) / 49 in Klein and Fricke 1890.
%H A121593 Seiichi Manyama, <a href="/A121593/b121593.txt">Table of n, a(n) for n = 1..10000</a>
%H A121593 Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
%H A121593 F. Klein and R. Fricke, <a href="https://archive.org/stream/vorlesungenuberd028637mbp#page/n767/mode/2up">Vorlesungen über die Theorie der elliptischen Modulfunctionen</a>, Teubner, Leipzig, 1890, Vol. 1, see p. 745, Eq. (3).
%F A121593 Euler transform of period 7 sequence [4, 4, 4, 4, 4, 4, 0, ...].
%F A121593 G.f.: x * (Product_{k>0} (1 - x^(7*k)) / (1 - x^k))^4.
%F A121593 G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u + v) * (u - v)^2 - u*v * (1 + 7*u) * (1 + 7*v).
%F A121593 G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^-2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030181. - _Michael Somos_, Jan 02 2015
%F A121593 G.f. A(q) satisfies j(q) = f(49 * A(q)) where f(x) := (x^2 + 13*x + 49) * (x^2 + 5*x + 1)^3 / x. - _Michael Somos_, Jan 02 2015
%F A121593 Convolution inverse of A030181. - _Michael Somos_, Jan 02 2015
%F A121593 a(n) ~ exp(4*Pi*sqrt(n/7)) / (49 * sqrt(2) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%F A121593 a(1) = 1, a(n) = (4/(n-1))*Sum_{k=1..n-1} A113957(k)*a(n-k) for n > 1. - _Seiichi Manyama_, Apr 01 2017
%e A121593 G.f. = q + 4*q^2 + 14*q^3 + 40*q^4 + 105*q^5 + 252*q^6 + 574*q^7 + ...
%t A121593 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^7] / QPochhammer[ q])^4, {q, 0, n}]; (* _Michael Somos_, Jan 02 2015 *)
%t A121593 nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(7*k)) / (1 - x^k))^4, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%t A121593 eta[q_]:=q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^7] / eta[q])^4, {q, 0, n}]; Table[a[n], {n, 4, 35}] (* _Vincenzo Librandi_, Oct 18 2018 *)
%o A121593 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^7 + A) / eta(x + A))^4, n))};
%Y A121593 Cf. A030181.
%K A121593 nonn
%O A121593 1,2
%A A121593 _Michael Somos_, Aug 09 2006