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 A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function. 37
 1, 3, 0, -6, -3, 0, 0, 6, 0, -6, 0, 0, 6, 6, 0, 0, -3, 0, 0, 6, 0, -12, 0, 0, 0, 3, 0, -6, -6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, -12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 6, 0, -12, -3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, -6, -6, 0, 0, 6, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Zagier (2009) denotes the g.f. as f(z) in Case B which is associated with F(t) the g.f. of A006077. REFERENCES D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of f(q)^3 / f(q^3) in powers of q where f() is a Ramanujan theta function. Expansion of 2*b(q^4) - b(q) = b(q^2)^3 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function. Expansion of eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^3 in powers of q. Euler transform of period 12 sequence [ 3, -6, 2, -3, 3, -4, 3, -3, 2, -6, 3, -2, ...]. Moebius transform is period 36 sequence [ 3, -3, -9, -3, -3, 9, 3, 3, 0, 3, -3, 9, 3, -3, 9, -3, -3, 0, 3, 3, -9, 3, -3, -9, 3, -3, 0, -3, -3, -9, 3, 3, 9, 3, -3, 0, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227696. G.f.: f(q) = F(t(q)) where F() is the g.f. of A006077 and t() is the g.f. of A227454. G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^(3*k)). a(3*n + 2) = a(4*n + 2) = 0. a(n) = (-1)^n * A005928(n) = (-1)^(((n+1) mod 6 ) > 3) * A113062(n). A113062(n) = |a(n)|. a(3*n) = A180318(n). a(2*n + 1) = 3 * A123530(n). a(4*n) = A005928(n). EXAMPLE G.f. = 1 + 3*q - 6*q^3 - 3*q^4 + 6*q^7 - 6*q^9 + 6*q^12 + 6*q^13 - 3*q^16 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^3 / QPochhammer[ -q^3], {q, 0, n}] PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^3, n))} CROSSREFS Cf. A005928, A006077, A113062, A180318, A227454, A227696. The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.) Sequence in context: A071126 A077187 A011079 * A005928 A113062 A259659 Adjacent sequences:  A226532 A226533 A226534 * A226536 A226537 A226538 KEYWORD sign AUTHOR Michael Somos, Sep 22 2013 STATUS approved

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Last modified October 18 05:14 EDT 2019. Contains 328145 sequences. (Running on oeis4.)