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 A215690 Expansion of a(q) / b(q) in powers of q where a(), b() are cubic AGM theta functions. 4

%I

%S 1,9,27,81,198,459,972,1989,3861,7290,13284,23679,41148,70218,117504,

%T 193671,314262,503415,796068,1244988,1925910,2950668,4478328,6739497,

%U 10059228,14901471,21914442,32011119,46456272,67010679,96093864,137039922,194395221

%N Expansion of a(q) / b(q) in powers of q where a(), b() are cubic AGM theta functions.

%D O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See u, page 1.

%H Seiichi Manyama, <a href="/A215690/b215690.txt">Table of n, a(n) for n = 0..10000</a>

%H J. M. Borwein, P. B. Borwein and F. Garvan, <a href="http://dx.doi.org/10.1090/S0002-9947-1994-1243610-6">Some Cubic Modular Identities of Ramanujan</a>, Trans. Amer. Math. Soc. 343 (1994), 35-47.

%F Expansion of 1 + 3 * c(q^3) / b(q) in powers of q where b(), c() are cubic AGM theta functions.

%F Expansion of 1 + 9 * (eta(q^9) / eta(q))^3 in powers of q.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u * v - 1)^3 - (u^3 - 1) * (v^3 - 1).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u + 2)^3 - 9 * v^3 * (1 + u + u^2).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 + 2) * (u2 + 2) - 3 * (1 + u1 + u2) * u3*u6.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058091.

%F G.f.: 1 + 9 * x * (Product_{k>0} (1 - x^(9*k)) / (1 - x^k))^3.

%F a(n) = 9 * A121589(n) unless n=0.

%F Convolution inverse is A115784. Convolution with A005928 is A004016.

%F a(n) ~ exp(4*Pi*sqrt(n)/3) / (3 * sqrt(6) * n^(3/4)). - _Vaclav Kotesovec_, Nov 14 2015

%e 1 + 9*q + 27*q^2 + 81*q^3 + 198*q^4 + 459*q^5 + 972*q^6 + 1989*q^7 + 3861*q^8 + ...

%t QP = QPochhammer; s = 1 + 9*q*(QP[q^9]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 + 9 * x * (eta(x^9 + A) / eta(x + A))^3, n))}

%Y Cf. A004016, A005928, A058091, A115784, A121589.

%K nonn

%O 0,2

%A _Michael Somos_, Aug 20 2012

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Last modified June 12 08:13 EDT 2021. Contains 344943 sequences. (Running on oeis4.)