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%I #19 Mar 12 2021 22:24:47
%S 1,-3,9,-22,51,-108,221,-429,810,-1476,2631,-4572,7802,-13056,21519,
%T -34918,55935,-88452,138332,-213990,327852,-497592,748833,-1117692,
%U 1655719,-2434938,3556791,-5161808,7445631,-10677096,15226658,-21599469,30485268,-42817788
%N Expansion of q * (f(q^9) / f(q))^3 in powers of q where f() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Zagier (2009) denotes the g.f. as t(z) in Case B which is associated with F(t) the g.f. of A006077.
%D 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 Seiichi Manyama, <a href="/A227454/b227454.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H 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 Expansion of c(-q^3) / (-3 * b(-q)) in powers of q where b(), c() are cubic AGM theta functions.
%F Expansion of (eta(q) * eta(q^4) * eta(q^18)^3 / (eta(q^2)^3 * eta(q^9) * eta(q^36)))^3 in powers of q.
%F Euler transform of period 36 sequence [ -3, 6, -3, 3, -3, 6, -3, 3, 0, 6, -3, 3, -3, 6, -3, 3, -3, 0, -3, 3, -3, 6, -3, 3, -3, 6, 0, 3, -3, 6, -3, 3, -3, 6, -3, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227498.
%F G.f. t(q) satisfies f(q) = F(t(q)) where F() is the g.f. of A006077 and f() is the g.f. of A226535
%F G.f.: x * (Product_{k>0} (1 - (-x)^(9*k)) / (1 - (-x)^k))^3.
%F a(n) = -(-1)^n * A121589(n).
%e G.f. = q - 3*q^2 + 9*q^3 - 22*q^4 + 51*q^5 - 108*q^6 + 221*q^7 - 429*q^8 + ...
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q^9] / QPochhammer[ -q])^3, {q, 0, n}]
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^3 / (eta(x^2 + A)^3 * eta(x^9 + A) * eta(x^36 + A)))^3, n))}
%Y Cf. A006077, A121589, A226535, A227498.
%Y 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 sign
%O 1,2
%A _Michael Somos_, Sep 22 2013
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