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 A131985 Expansion of (eta(q^3)^2 / (eta(q) * eta(q^9)))^6 in powers of q. 3
 1, 6, 27, 86, 243, 594, 1370, 2916, 5967, 11586, 21870, 39852, 71052, 123444, 210654, 352480, 581013, 942786, 1510254, 2388204, 3734964, 5777788, 8852004, 13434984, 20218395, 30177684, 44704413, 65743348, 96033357, 139368816, 201032186, 288281592, 411119766 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,2 COMMENTS In Berndt and Chan (1999) denoted by h(q) in Theorem 3.1. - Michael Somos, Oct 20 2013 LINKS Seiichi Manyama, Table of n, a(n) for n = -1..10000 B. C. Berndt and H. H. Chan, Ramanujan and the Modular j-invariant, Canad. Math. Bull., 42 (1999), 427-440. FORMULA Euler transform of period 9 sequence [ 6, 6, -6, 6, 6, -6, 6, 6, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+v)^3 + u*v*(27 + 9*(u+v) - u*v). G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + w^2 + 12*v^2 + u*w - v^2*(u+w) + 12*v*(u+w) + 27*v. G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = f(t) where q = exp(2 Pi i t). G.f.: (1/x) * (Product_{k>0} (1 - x^(3*k))^2 / ((1 - x^k) * (1 - x^(9*k))))^6. a(n) = A007266(n) = A045491(n) unless n=0. a(n) = A131986(n) + 27 * A121589(n) unless n=0. - Michael Somos, Oct 20 2013 Convolution inverse of A121592. - Michael Somos, Oct 20 2013 a(n) ~ exp(4*Pi*sqrt(n)/3) / (sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015 EXAMPLE G.f. = 1/q + 6 + 27*q + 86*q^2 + 243*q^3 + 594*q^4 + 1370*q^5 + 2916*q^6 +... MATHEMATICA a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^3]^2 / (QPochhammer[ q] QPochhammer[ q^9]))^6, {q, 0, n}]; (* Michael Somos, Oct 20 2013 *) PROG (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A)^2 / (eta(x + A) * eta(x^9 + A)))^6, n))}; CROSSREFS Cf. A007266, A045491, A121589, A131986. Sequence in context: A124089 A250283 A100188 * A125196 A100189 A052267 Adjacent sequences:  A131982 A131983 A131984 * A131986 A131987 A131988 KEYWORD nonn AUTHOR Michael Somos, Aug 04 2007 STATUS approved

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Last modified April 3 17:03 EDT 2020. Contains 333197 sequences. (Running on oeis4.)