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A121591 Expansion of (eta(q^5) / eta(q))^6 in powers of q. 8
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 13 13:48 EDT 2020. Contains 335688 sequences. (Running on oeis4.)