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

Author

Michael Somos, Aug 09 2006

Keywords

• nonn

Examples

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

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.

Crossrefs

Cf. A035959, A106248.

Programs

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 *)

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))};

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