OFFSET
-1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..10000
FORMULA
Expansion of (eta(q^3) * eta(q^7) / (eta(q) * eta(q^21)))^2 in powers of q.
Euler transform of period 21 sequence [ 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u*v + 3) - (u+v) * (u^2 + 3 * u*v + v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (21 t)) = f(t) where q = exp(2 Pi i t).
G.f.: 1/x * (Product_{k>0} (1 - x^(3*k)) * (1 - x^(7*k)) / ((1 - x^k) * (1 - x^(21*k))))^2.
a(n) = A058566(n) unless n=0.
a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
EXAMPLE
1/q + 2 + 5*q + 8*q^2 + 16*q^3 + 26*q^4 + 44*q^5 + 66*q^6 + 104*q^7 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[((1 - x^(3*k)) * (1 - x^(7*k)) / ((1 - x^k) * (1 - x^(21*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 06 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q^3] *eta[q^7]/(eta[q]*eta[q^21]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 17 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^21 + A)))^2, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 22 2013
STATUS
approved