A232772 Expansion of (psi(x)^2 / (phi(-x) * phi(x^2)))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
1, 8, 30, 80, 197, 472, 1046, 2160, 4306, 8360, 15712, 28656, 51127, 89552, 153926, 259904, 432336, 709728, 1150142, 1841200, 2915546, 4570904, 7097622, 10921184, 16664073, 25228176, 37907758, 56553936, 83806768, 123405752, 180611558, 262799248, 380275604
Offset: 0
Keywords
Examples
G.f. = 1 + 8*x + 30*x^2 + 80*x^3 + 197*x^4 + 472*x^5 + 1046*x^6 + 2160*x^7 + ... G.f. = q + 8*q^3 + 30*q^5 + 80*q^7 + 197*q^9 + 472*q^11 + 1046*q^13 + ...
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
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^4 / (16 q^(1/2)) / (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}] nmax=60; CoefficientList[Series[Product[((1-x^k)^3 * (1+x^k)^7 * (1+x^(4*k))^2 / (1-x^(4*k))^3)^2,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *) -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 * eta(x^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5))^2, n))}
Formula
Expansion of q^(-1/2) * (eta(q^2)^7 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5))^2 in powers of q.
Euler transform of period 8 sequence [ 8, -6, 8, 4, 8, -6, 8, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 g(t) where q = eqp(2 Pi i t) and g() is the g.f. of A233458.
a(n) ~ exp(sqrt(2*n)*Pi) / (2^(17/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
Comments