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
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
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 09 2006
STATUS
approved