A121597 Expansion of (eta(q^13) / eta(q))^2 in powers of q.

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 752, 1165, 1768, 2661, 3946, 5802, 8430, 12158, 17360, 24622, 34632, 48410, 67188, 92731, 127182, 173546, 235508, 318098, 427536, 572168, 762318, 1011660, 1337136, 1760876, 2310338, 3021008, 3936848

Offset

1,2

Comments

The g.f. is an eta-quotient and a modular function. - Michael Somos, Feb 19 2018

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.

Wikipedia, Modular functions

Formula

Euler transform of period 13 sequence [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, ...].

G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) where f(u, v) = u^3 + v^3 - u*v - 4 * u*v * (u + v) - 13 * u^2*v^2.

G.f.: x * (Product_{k>0} (1 - x^(13*k)) / (1 - x^k))^2.

G.f. A(x) satisfies: 0 = f(A(x), A(x^3)) where f(u, v) = (u^2 - u*v + v^2)^2 - u*v * (1 + 6*u + 13*u^2) * (1 + 6*v + 13*v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (13 t)) = (1/13) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A133099.

Convolution inverse of A133099.

a(n) ~ exp(4*Pi*sqrt(n/13)) / (sqrt(2) * 13^(5/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

a(1) = 1, a(n) = (2/(n-1))*Sum_{k=1..n-1} A284587(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

A034318(n) = A133099(n) + 13 * a(n). - Michael Somos, Feb 19 2018

Example

G.f. = q + 2*q^2 + 5*q^3 + 10*q^4 + 20*q^5 + 36*q^6 + 65*q^7 + 110*q^8 + ...

Mathematica

nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(13*k)) / (1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)
QP = QPochhammer; s = (QP[q^13]/QP[q])^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^13] / QPochhammer[ q])^2, {q, 0, n}]; (* Michael Somos, Feb 19 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^13 + A) / eta(x + A))^2, n))};

Crossrefs

Cf. A034318, A133099.

Sequence in context: A103927 A103928 A103929 * A000712 A032442 A327293

Adjacent sequences: A121594 A121595 A121596 * A121598 A121599 A121600

Keyword

nonn

Author

Michael Somos, Aug 09 2006

Status

approved