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A121593 Expansion of (eta(q^7) / eta(q))^4 in powers of q. 3
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 (list; graph; refs; listen; history; text; internal format)
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 ├╝ber 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

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Last modified May 7 20:36 EDT 2021. Contains 343652 sequences. (Running on oeis4.)