OFFSET
0,2
COMMENTS
LINKS
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 q^(-3/8) * eta(q^2) * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 0, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 + x^k + x^(2*k)) * (1 + x^(6*k)).
a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(11/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + ...
G.f. = q^3 + 2*q^11 + 4*q^19 + 7*q^27 + 12*q^35 + 20*q^43 + 32*q^51 + 50*q^59 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(12*k))/( (1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 2, Pi/4, x^(3/2)] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 13 2007
STATUS
approved