A279320 Expansion of chi(-x^4) * psi(x^6) / phi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
1, 2, 4, 8, 15, 26, 45, 74, 119, 188, 291, 442, 664, 982, 1435, 2076, 2972, 4214, 5929, 8272, 11457, 15762, 21543, 29264, 39532, 53110, 70988, 94430, 125033, 164826, 216388, 282940, 368552, 478326, 618621, 797376, 1024485, 1312184, 1675657, 2133664, 2709307
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 45*x^6 + 74*x^7 + ... G.f. = q^11 + 2*q^23 + 4*q^35 + 8*q^47 + 15*q^59 + 26*q^71 + 45*q^83 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..2500
- 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
Crossrefs
Cf. A131945.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ x^(-3/4)/2 QPochhammer[ -x^4, x^4] EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 4, 0, x], {x, 0, n}]; a[ n_] := SeriesCoefficient[ 2^(-3/2)/x EllipticTheta[ 2, Pi/4, x] EllipticTheta[ 2, 0, x^3] / QPochhammer[ x]^2, {x, 0, n}]; nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(4*k)) * (1+x^(6*k)) * (1-x^(12*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 10 2016 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};
Formula
Expansion of f(-x^2, -x^10) * f(-x^8) / f(-x)^2 in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of q^(-11/12) * eta(q^2) * eta(q^8) * eta(q^12)^2 / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 24 sequence [ 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 0, ...].
a(n) ~ exp(sqrt(13*n/3)*Pi/2) * 13^(1/4) / (16*sqrt(2)*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, Dec 10 2016
Comments