OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (eta(q^2) * eta(q^13) / (eta(q) * eta(q^26)))^2 in powers of q.
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, ...].
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).
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).
G.f. is a period 1 Fourier series which satisfies f(-1 / (26 t)) = f(t) where q = exp(2 Pi i t).
G.f.: (1/x)* (Product_{k>0} P(x^k))^-2 where P(x) is the 26th cyclotomic polynomial of degree 12.
a(n) = A058597(n) unless n = 0.
a(n) ~ exp(2*Pi*sqrt(2*n/13)) / (2^(3/4) * 13^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
EXAMPLE
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 + ...
MATHEMATICA
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 *)
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 *)
PROG
(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))};
(Magma) A := Basis( CuspForms( Gamma0(26), 2), 46); B<q> := A[1] / A[2]; B; /* Michael Somos, Nov 30 2014 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Mar 06 2007
STATUS
approved