A260599 Expansion of psi(x^4) / chi(-x)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
1, 2, 3, 6, 10, 16, 25, 38, 55, 80, 115, 160, 223, 306, 415, 560, 747, 988, 1301, 1700, 2206, 2850, 3661, 4676, 5950, 7536, 9500, 11936, 14936, 18620, 23141, 28662, 35386, 43566, 53480, 65466, 79937, 97356, 118277, 143370, 173391, 209232, 251966, 302806
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + ... G.f. = q^7 + 2*q^19 + 3*q^31 + 6*q^43 + 10*q^55 + 16*q^67 + 25*q^79 + ...
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
Crossrefs
Cf. A260574.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}]; a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^4 / (QPochhammer[ x]^2 EllipticTheta[ 3, 0, x^2]), {x, 0, n}]; nmax=60; CoefficientList[Series[Product[(1+x^k)^2 * (1-x^(8*k))^2 / (1-x^(4*k)),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))};
Formula
Expansion of f(-x^4)^4 / (f(-x)^2 * phi(x^2)) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-7/12) * eta(q^2)^2 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.
Euler transform of period 8 sequence [ 2, 0, 2, 1, 2, 0, 2, -1, ...].
2 * a(n) = A260574(4*n + 2).
a(n) ~ exp(sqrt(2*n/3)*Pi) / (8*sqrt(2*n)). - Vaclav Kotesovec, Oct 14 2015
Comments