A121591 Expansion of (eta(q^5) / eta(q))^6 in powers of q.

1, 6, 27, 98, 315, 912, 2456, 6210, 14937, 34390, 76317, 163896, 342062, 695736, 1382880, 2691586, 5139906, 9644622, 17808040, 32393370, 58113312, 102914152, 180062622, 311488920, 533124225, 903324372, 1516110165, 2521780688, 4158863310, 6803237280, 11043320922, 17794350786

Offset

1,2

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.

Formula

Euler transform of period 5 sequence [6, 6, 6, 6, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (1 + 125 * u*v) - (u+v) * (u^2 - 13 * u*v + v^2). - Michael Somos, May 22 2013

G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 1/125 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106248. - Michael Somos, May 22 2013

G.f.: x * (Product_{k>0} (1 - x^(5*k)) / (1 - x^k))^6.

Convolution inverse of A106248, 6th power of A035959. - Michael Somos, Aug 09 2015

a(n) ~ exp(4*Pi*sqrt(n/5)) / (125 * sqrt(2) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

a(1) = 1, a(n) = (6/(n-1))*Sum_{k=1..n-1} A116073(k)*a(n-k) for n > 1. - Seiichi Manyama, Mar 31 2017

Example

G.f. = q + 6*q^2 + 27*q^3 + 98*q^4 + 315*q^5 + 912*q^6 + 2456*q^7 + 6210*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^5] / QPochhammer[ q])^6, {q, 0, n}]; (* Michael Somos, May 22 2013 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(5*k)) / (1 - x^k))^6, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^5 + A) / eta(x + A))^6, n))};

Crossrefs

Cf. A035959, A106248.

Sequence in context: A278357 A001940 A320049 * A071734 A160507 A182821

Adjacent sequences: A121588 A121589 A121590 * A121592 A121593 A121594

Keyword

nonn

Author

Michael Somos, Aug 09 2006

Status

approved