OFFSET
0,2
COMMENTS
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 7th equation.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of psi(x^6) * phi(-x^2) / f(-x)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q^2)^2 * eta(q^12)^2 / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, 0, 2, 1, 2, 1, 2, 1, 2, 0, 2, 0, ...].
-2 * a(n) = A262967(3*n + 2).
a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015
G.f.: Sum_{k>=0} x^k * (Product_{i=1..k} 1 + x^(2*i)) / Product_{i=1..2*k+1} 1 - x^i). [Ramanujan] - Michael Somos, Nov 18 2015
EXAMPLE
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 26*x^6 + 40*x^7 + ...
G.f. = q^2 + 2*q^5 + 3*q^8 + 6*q^11 + 10*q^14 + 16*q^17 + 26*q^20 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^3] EllipticTheta[ 4, 0, x^2] / (2 x^(3/4) QPochhammer[ x]^2), {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^(12*k)) * (1+x^(6*k)) * (1+x^(2*k-1)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)^2*eta(q^12)^2/(eta(q)^2*eta(q^4)*eta(q^6))) \\ Altug Alkan, Mar 19 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 06 2015
STATUS
approved