OFFSET
0,4
COMMENTS
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 q^(-5/6) * eta(q^6)^2 * eta(q^24)^2 / (eta(q) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, -1, ...].
-2 * a(n) = A260574(4*n + 3).
a(n) ~ exp(sqrt(2*n/3)*Pi) / (24*sqrt(2*n)). - Vaclav Kotesovec, Oct 14 2015
EXAMPLE
G.f. = x + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 20*x^8 + ...
G.f. = q^11 + q^17 + 2*q^23 + 4*q^29 + 6*q^35 + 9*q^41 + 14*q^47 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^6] / ( 4 x^(7/8) QPochhammer[ x]), {x, 0, n}];
nmax=60; CoefficientList[Series[x*Product[(1-x^(6*k)) * (1-x^(24*k)) * (1+x^(3*k)) * (1+x^(12*k)) / ((1-x^k)), {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^6 + A)^2 * eta(x^24 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); concat(0, Vec(eta(q^6)^2*eta(q^24)^2 / (eta(q)*eta(q^3)*eta(q^12)))) \\ Altug Alkan, Mar 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 29 2015
STATUS
approved