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 A215690 Expansion of a(q) / b(q) in powers of q where a(), b() are cubic AGM theta functions. 4
 1, 9, 27, 81, 198, 459, 972, 1989, 3861, 7290, 13284, 23679, 41148, 70218, 117504, 193671, 314262, 503415, 796068, 1244988, 1925910, 2950668, 4478328, 6739497, 10059228, 14901471, 21914442, 32011119, 46456272, 67010679, 96093864, 137039922, 194395221 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES 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. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), 35-47. FORMULA Expansion of 1 + 3 * c(q^3) / b(q) in powers of q where b(), c() are cubic AGM theta functions. Expansion of 1 + 9 * (eta(q^9) / eta(q))^3 in powers of q. 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). 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). 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. 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. G.f.: 1 + 9 * x * (Product_{k>0} (1 - x^(9*k)) / (1 - x^k))^3. a(n) = 9 * A121589(n) unless n=0. Convolution inverse is A115784. Convolution with A005928 is A004016. a(n) ~ exp(4*Pi*sqrt(n)/3) / (3 * sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Nov 14 2015 EXAMPLE 1 + 9*q + 27*q^2 + 81*q^3 + 198*q^4 + 459*q^5 + 972*q^6 + 1989*q^7 + 3861*q^8 + ... MATHEMATICA 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 *) PROG (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))} CROSSREFS Cf. A004016, A005928, A058091, A115784, A121589. Sequence in context: A370870 A053702 A036314 * A255623 A036317 A053762 Adjacent sequences: A215687 A215688 A215689 * A215691 A215692 A215693 KEYWORD nonn AUTHOR Michael Somos, Aug 20 2012 STATUS approved

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Last modified August 6 04:44 EDT 2024. Contains 374957 sequences. (Running on oeis4.)