A121597 - OEIS

%I #25 Oct 18 2018 03:06:24

%S 1,2,5,10,20,36,65,110,185,300,481,752,1165,1768,2661,3946,5802,8430,

%T 12158,17360,24622,34632,48410,67188,92731,127182,173546,235508,

%U 318098,427536,572168,762318,1011660,1337136,1760876,2310338,3021008,3936848

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

%C The g.f. is an eta-quotient and a modular function. - _Michael Somos_, Feb 19 2018

%H Seiichi Manyama, <a href="/A121597/b121597.txt">Table of n, a(n) for n = 1..10000</a>

%H Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Modular_form#Modular_functions">Modular functions</a>

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

%F 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.

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

%F 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).

%F 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.

%F Convolution inverse of A133099.

%F a(n) ~ exp(4*Pi*sqrt(n/13)) / (sqrt(2) * 13^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015

%F 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

%F A034318(n) = A133099(n) + 13 * a(n). - _Michael Somos_, Feb 19 2018

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

%t 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 *)

%t QP = QPochhammer; s = (QP[q^13]/QP[q])^2 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q^13] / QPochhammer[ q])^2, {q, 0, n}]; (* _Michael Somos_, Feb 19 2018 *)

%o (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))};

%Y Cf. A034318, A133099.

%K nonn

%O 1,2

%A _Michael Somos_, Aug 09 2006