A137829 Expansion of psi(q^2) / f(-q)^2 in powers of q where psi(), f() are Ramanujan theta functions.
1, 2, 6, 12, 25, 46, 86, 148, 255, 420, 686, 1088, 1712, 2634, 4020, 6036, 8988, 13214, 19282, 27840, 39923, 56750, 80160, 112384, 156660, 216958, 298894, 409420, 558119, 756950, 1022090, 1373760, 1838932, 2451366, 3255480, 4306920, 5678104, 7459634, 9768386
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 25*x^4 + 46*x^5 + 86*x^6 + 148*x^7 + ... G.f. = q + 2*q^7 + 6*q^13 + 12*q^19 + 25*q^25 + 46*q^31 + 86*q^37 + 148*q^43 + ...
References
- Lusztig, G., Irreducible representation of finite classical groups, Inventiones Math., 43 (1977), 125-175. See p. 135.
- Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table II.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Márton Balázs, Dan Fretwell, and Jessica Jay, Interacting Particle Systems and Jacobi style identities, arXiv:2011.05006 [math.PR], 2020.
- 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
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]^2), {x, 0, n}]; (* Michael Somos, Oct 04 2015 *) nmax=60; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k))^2 / (1-x^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)^2 / (eta(x + A)^2 * eta(x^2 + A)), n))};
Formula
Expansion of q^(-1/6) * eta(q^4)^2 / (eta(q)^2 * eta(q^2)) in powers of q.
Euler transform of period 4 sequence [ 2, 3, 2, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 96^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137830.
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k).
2 * a(n) = A137828(4*n + 1).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Oct 13 2015
Comments