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Expansion of q^(-1) * (chi(-q^13) / chi(-q))^2 in powers of q where chi() is a Ramanujan theta function.
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%I #19 Sep 08 2022 08:45:30

%S 1,2,3,6,9,14,22,32,46,66,93,128,176,236,315,420,550,718,932,1198,

%T 1534,1956,2476,3120,3919,4896,6095,7562,9341,11504,14126,17284,21090,

%U 25666,31140,37692,45515,54818,65878,79000,94523,112872,134522,160004

%N Expansion of q^(-1) * (chi(-q^13) / chi(-q))^2 in powers of q where chi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A128518/b128518.txt">Table of n, a(n) for n = -1..1000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (eta(q^2) * eta(q^13) / (eta(q) * eta(q^26)))^2 in powers of q.

%F Euler transform of period 26 sequence [ 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (w^2 - v) - u*w * (4*(1+v^2) - 4*v).

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (26 t)) = f(t) where q = exp(2 Pi i t).

%F G.f.: (1/x)* (Product_{k>0} P(x^k))^-2 where P(x) is the 26th cyclotomic polynomial of degree 12.

%F a(n) = A058597(n) unless n = 0.

%F a(n) ~ exp(2*Pi*sqrt(2*n/13)) / (2^(3/4) * 13^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015

%e G.f. = 1/q + 2 + 3*q + 6*q^2 + 9*q^3 + 14*q^4 + 22*q^5 + 32*q^6 + 46*q^7 + ...

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^2] QPochhammer[ q^13] / (QPochhammer[ q] QPochhammer[ q^26]))^2, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t nmax=60; CoefficientList[Series[Product[((1+x^k) / (1+x^(13*k)))^2,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^13 + A) / (eta(x + A) * eta(x^26 + A)))^2, n))};

%o (Magma) A := Basis( CuspForms( Gamma0(26), 2), 46); B<q> := A[1] / A[2]; B; /* _Michael Somos_, Nov 30 2014 */

%Y Cf. A058597.

%K nonn

%O -1,2

%A _Michael Somos_, Mar 06 2007