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

1, 4, 14, 40, 105, 252, 574, 1236, 2564, 5124, 9948, 18788, 34685, 62664, 111132, 193672, 332325, 561996, 937958, 1546132, 2519825, 4062888, 6486008, 10257324, 16079389, 24996636, 38555216, 59025820, 89728900, 135486960, 203274344

Offset

1,2

Comments

G.f. A(q) is denoted by tau(q) / 49 in Klein and Fricke 1890.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

F. Klein and R. Fricke, Vorlesungen über die Theorie der elliptischen Modulfunctionen, Teubner, Leipzig, 1890, Vol. 1, see p. 745, Eq. (3).

Formula

Euler transform of period 7 sequence [4, 4, 4, 4, 4, 4, 0, ...].

G.f.: x * (Product_{k>0} (1 - x^(7*k)) / (1 - x^k))^4.

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).

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

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

Convolution inverse of A030181. - Michael Somos, Jan 02 2015

a(n) ~ exp(4*Pi*sqrt(n/7)) / (49 * sqrt(2) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

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

Example

G.f. = q + 4*q^2 + 14*q^3 + 40*q^4 + 105*q^5 + 252*q^6 + 574*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^7] / QPochhammer[ q])^4, {q, 0, n}]; (* Michael Somos, Jan 02 2015 *)
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 *)
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 *)

Prog

(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))};

Crossrefs

Cf. A030181.

Sequence in context: A274327 A160463 A278680 * A160527 A023003 A001872

Adjacent sequences: A121590 A121591 A121592 * A121594 A121595 A121596

Keyword

nonn

Author

Michael Somos, Aug 09 2006

Status

approved