OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1) * ((chi(-q^3) * chi(-q^15)) / (chi(-q) * chi(-q^5)))^2 in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q^2) * eta(q^3) * eta(q^10) * eta(q^15) / (eta(q) * eta(q^5) * eta(q^6) * eta(q^30)))^2 in powers of q.
Euler transform of period 30 sequence [ 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 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) - 4*u*v * (v - 1)^2. - Michael Somos, Jun 09 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = f(t) where q = exp(2 Pi i t).
G.f.: (1/x) * (Product_{k>0} (1 + x^k) * (1 + x^(5*k)) / ((1 + x^(3*k)) * (1 + x^(15*k))))^2.
a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
EXAMPLE
1/q + 2 + 3*q + 4*q^2 + 5*q^3 + 10*q^4 + 15*q^5 + 22*q^6 + 29*q^7 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[((1 + x^k) * (1 + x^(5*k)) / ((1 + x^(3*k)) * (1 + x^(15*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^2] *eta[q^3]*eta[q^10]*eta[q^15]/(eta[q]*eta[q^5]*eta[q^6]*eta[q^30]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 06 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^10 + A) * eta(x^15 + A) / (eta(x + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^30 + A)))^2, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Feb 02 2012
STATUS
approved