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A137829 Expansion of psi(q^2) / f(-q)^2 in powers of q where psi(), f() are Ramanujan theta functions.

%I A137829 #26 Feb 16 2025 08:33:07
%S A137829 1,2,6,12,25,46,86,148,255,420,686,1088,1712,2634,4020,6036,8988,
%T A137829 13214,19282,27840,39923,56750,80160,112384,156660,216958,298894,
%U A137829 409420,558119,756950,1022090,1373760,1838932,2451366,3255480,4306920,5678104,7459634,9768386
%N A137829 Expansion of psi(q^2) / f(-q)^2 in powers of q where psi(), f() are Ramanujan theta functions.
%C A137829 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A137829 Moreno (preprint) calls this |Phi_n|. - _N. J. A. Sloane_, Sep 01 2018
%D A137829 Lusztig, G., Irreducible representation of finite classical groups, Inventiones Math., 43 (1977), 125-175. See p. 135.
%D A137829 Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table II.
%H A137829 G. C. Greubel, <a href="/A137829/b137829.txt">Table of n, a(n) for n = 0..1000</a>
%H A137829 Márton Balázs, Dan Fretwell, and Jessica Jay, <a href="https://arxiv.org/abs/2011.05006">Interacting Particle Systems and Jacobi style identities</a>, arXiv:2011.05006 [math.PR], 2020.
%H A137829 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%H A137829 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A137829 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A137829 Expansion of q^(-1/6) * eta(q^4)^2 / (eta(q)^2 * eta(q^2)) in powers of q.
%F A137829 Euler transform of period 4 sequence [ 2, 3, 2, 1, ...].
%F A137829 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.
%F A137829 G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k).
%F A137829 2 * a(n) = A137828(4*n + 1).
%F A137829 a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - _Vaclav Kotesovec_, Oct 13 2015
%e A137829 G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 25*x^4 + 46*x^5 + 86*x^6 + 148*x^7 + ...
%e A137829 G.f. = q + 2*q^7 + 6*q^13 + 12*q^19 + 25*q^25 + 46*q^31 + 86*q^37 + 148*q^43 + ...
%t A137829 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]^2), {x, 0, n}]; (* _Michael Somos_, Oct 04 2015 *)
%t A137829 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 *)
%o A137829 (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))};
%Y A137829 Cf. A137828, A137830.
%Y A137829 Cf. Half of A201078, which gives another application.
%K A137829 nonn
%O A137829 0,2
%A A137829 _Michael Somos_, Feb 12 2008