A145722 Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.
1, 1, 3, 4, 8, 12, 21, 30, 48, 68, 102, 143, 207, 284, 400, 542, 744, 996, 1344, 1776, 2361, 3088, 4050, 5248, 6808, 8742, 11232, 14310, 18224, 23052, 29133, 36601, 45936, 57360, 71528, 88812, 110110, 135990, 167704, 206108, 252912, 309408
Offset: 0
Keywords
Examples
G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 30*x^7 + 48*x^8 + ... G.f. = q + q^5 + 3*q^9 + 4*q^13 + 8*q^17 + 12*q^21 + 21*q^25 + 30*q^29 + ...
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], Sep 30 2015
- MathOverflow, Up to 2000..., 2016.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Crossrefs
Cf. A036026.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2]^2, {x, 0, n}]; (* Michael Somos, Aug 26 2015 *) nmax=60; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(10*k)) * (1+x^(5*k)) / ((1-x^k) * (1 + x^(10*k))),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};
Formula
Expansion of q^(-1/4) * eta(q^4) * eta(q^10)^3 / (eta(q) * eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 1, 2, 1, 1, 2, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 2, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(-1/2) g(t), where q = exp(2 Pi i t) and g() is the g.f. for A145723.
G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(10*k)) * (1 + x^(5*k)) / ((1 - x^k) * (1 + x^(10*k))).
a(n) = A036026(2*n).
a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
Comments