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A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

%I A227216 #38 Feb 16 2025 08:33:20
%S A227216 1,3,4,2,1,3,6,4,0,-1,4,6,4,2,2,2,3,4,2,0,1,6,8,2,0,3,6,0,-2,0,6,6,4,
%T A227216 4,2,4,3,4,0,-2,0,6,8,2,2,-1,6,4,2,1,4,6,4,2,0,6,0,0,0,0,4,6,8,2,1,2,
%U A227216 12,4,-2,-2,2,6,0,2,2,2,0,8,4,0,3,3,8,2
%N A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.
%C A227216 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A227216 Zagier (2009) refers to Case D corresponding to the Apery numbers (A005258).
%D A227216 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 A227216 Seiichi Manyama, <a href="/A227216/b227216.txt">Table of n, a(n) for n = 0..10000</a>
%H A227216 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A227216 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A227216 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 A227216 Expansion of f(-q)^2 * (f(-q^5) / f(-q, -q^4))^5 = f(-q^2, -q^3)^2 * (f(-q^5) / f(-q, -q^4))^3 in powers of q where f() is a Ramanujan theta function.
%F A227216 Euler transform of period 5 sequence [ 3, -2, -2, 3, -2, ...].
%F A227216 Moebius transform is period 5 sequence [ 3, 1, -1, -3, 0, ...]. - _Michael Somos_, Jun 10 2014
%F A227216 G.f. = g(t(q)) where g(), t() are the g.f. for A005258 and A078905.
%F A227216 G.f.: (Product_{k>0} (1 - x^k)^2) / (Product_{k>0} (1 - x^(5*k - 1)) * (1 - x^(5*k - 4)))^5.
%e A227216 G.f. = 1 + 3*q + 4*q^2 + 2*q^3 + q^4 + 3*q^5 + 6*q^6 + 4*q^7 - q^9 + ...
%t A227216 a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ Re[(3 - I) {1, I, -I, -1, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]];
%t A227216 a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 / (QPochhammer[ q, q^5] QPochhammer[ q^4, q^5])^5, {q, 0, n}]; (* _Michael Somos_, Jun 10 2014 *)
%o A227216 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, real( (3 - I) * [ 0, 1, I, -I, -1][ d%5 + 1])))};
%o A227216 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -3, 2, 2, -3][k%5 + 1], 1 + x * O(x^n)), n))};
%o A227216 (Sage) A = ModularForms( Gamma1(5), 1, prec=20) . basis(); A[0] + 3*A[1]; # _Michael Somos_, Jun 10 2014
%o A227216 (Magma) A := Basis( ModularForms( Gamma1(5), 1), 20); A[1] + 3*A[2]; /* _Michael Somos_, Jun 10 2014 */
%Y A227216 Cf. A005258, A078905, A229802.
%Y A227216 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 A227216 sign
%O A227216 0,2
%A A227216 _Michael Somos_, Sep 21 2013