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A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.

%I A226535 #42 Feb 16 2025 08:33:19
%S A226535 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,
%T A226535 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,
%U A226535 0,-12,-3,0,0,6,0,0,0,0,0,6,0,-6,-6,0,0,6,0
%N A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.
%C A226535 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A226535 Zagier (2009) denotes the g.f. as f(z) in Case B which is associated with F(t) the g.f. of A006077.
%D A226535 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.
%H A226535 Seiichi Manyama, <a href="/A226535/b226535.txt">Table of n, a(n) for n = 0..10000</a>
%H A226535 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226535 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A226535 D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf">Integral solutions of Apery-like recurrence equations</a>.
%F A226535 Expansion of f(q)^3 / f(q^3) in powers of q where f() is a Ramanujan theta function.
%F A226535 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.
%F A226535 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.
%F A226535 Euler transform of period 12 sequence [ 3, -6, 2, -3, 3, -4, 3, -3, 2, -6, 3, -2, ...].
%F A226535 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, ...].
%F A226535 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.
%F A226535 G.f.: f(q) = F(t(q)) where F() is the g.f. of A006077 and t() is the g.f. of A227454.
%F A226535 G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^(3*k)).
%F A226535 a(3*n + 2) = a(4*n + 2) = 0.
%F A226535 a(n) = (-1)^n * A005928(n) = (-1)^(((n+1) mod 6 ) > 3) * A113062(n). A113062(n) = |a(n)|.
%F A226535 a(3*n) = A180318(n). a(2*n + 1) = 3 * A123530(n). a(4*n) = A005928(n).
%e A226535 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 + ...
%t A226535 a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^3 / QPochhammer[ -q^3], {q, 0, n}]
%o A226535 (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))}
%Y A226535 Cf. A005928, A006077, A113062, A180318, A227454, A227696.
%Y A226535 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.)
%K A226535 sign
%O A226535 0,2
%A A226535 _Michael Somos_, Sep 22 2013