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A128638 Expansion of q * (psi(q^3)^3 / psi(q)) / (phi(-q)^3 / phi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.

%I A128638 #33 Feb 16 2025 08:33:05
%S A128638 1,5,19,61,174,455,1112,2573,5689,12102,24900,49759,96902,184408,
%T A128638 343722,628717,1130418,2000669,3489788,6005910,10207688,17147892,
%U A128638 28494120,46865519,76342903,123236446,197233723,313106264,493231830,771301986,1197743552,1847606573
%N A128638 Expansion of q * (psi(q^3)^3 / psi(q)) / (phi(-q)^3 / phi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
%C A128638 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A128638 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A128638 G. C. Greubel, <a href="/A128638/b128638.txt">Table of n, a(n) for n = 1..1000</a>
%H A128638 Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
%H A128638 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128638 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128638 Expansion of (1/3) * (c(q^2)^2 / c(q)) / (b(q)^2 / b(q^2)) in powers of q where b(), c() are cubic AGM theta functions.
%F A128638 Expansion of (eta(q^6) / eta(q))^5 * eta(q^2) / eta(q^3) in powers of q.
%F A128638 Euler transform of period 6 sequence [ 5, 4, 6, 4, 5, 0, ...].
%F A128638 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 + 8*u) * (1 + 9*v) - (u-v)^2.
%F A128638 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/72) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128632.
%F A128638 G.f.: x * Product_{k>0} ((1 - x^(6*k)) / (1-x^k))^5 * ((1 - x^(2*k)) / (1 - x^(3*k))).
%F A128638 8 * a(n) = A128639(n) unless n = 0. Convolution inverse of A128632.
%F A128638 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (72 * 2^(3/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%e A128638 G.f. = q + 5*q^2 + 19*q^3 + 61*q^4 + 174*q^5 + 455*q^6 + 1112*q^7 + ...
%t A128638 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^6] / QPochhammer[ q])^5 (QPochhammer[ q^2] / QPochhammer[ q^3]), {q, 0, n}]; (* _Michael Somos_, Jun 08 2015 *)
%t A128638 nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(6*k)) / (1-x^k))^5 * ((1 - x^(2*k)) / (1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o A128638 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) / eta(x + A))^5 * eta(x^2 + A) / eta(x^3 + A), n))};
%Y A128638 Cf. A128632, A128639.
%K A128638 nonn
%O A128638 1,2
%A A128638 _Michael Somos_, Mar 16 2007
%E A128638 Edited by _N. J. A. Sloane_, Apr 01 2008