%I #23 Sep 06 2018 02:41:43
%S 1,-5,9,-11,24,-45,50,-53,81,-120,120,-99,170,-250,216,-203,288,-405,
%T 362,-264,450,-600,528,-477,601,-850,729,-550,840,-1080,962,-821,1080,
%U -1440,1200,-891,1370,-1810,1530,-1272,1680,-2250,1850,-1320,1944,-2640,2208
%N Expansion of eta(q)^5 * eta(q^3) * eta(q^6)^4 / eta(q^2)^4 in powers of q.
%C Zagier (2009) writes "... associated to the weight 3 Eisenstein series g(z) = Sigma b(n)q^n = q - 5q^2 + 9q^3 - 11q^4 + ...".
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%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 Indranil Ghosh, <a href="/A214262/b214262.txt">Table of n, a(n) for n = 1..2000</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 (1/9) * c(q) * b(q)^2 * c(q^2) / b(q^2) = (c(q)^3 - 8*c(q^2)^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.
%F Euler transform of period 6 sequence [ -5, -1, -6, -1, -5, -6, ...]. - _Michael Somos_, Oct 06 2013
%F a(n) is multiplicative with a(3^e) = 9^e, a(2^e) = -(4^(e+1) + 9*(-1)^(e+1)) / 5, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e) = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 5 (mod 6).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 192^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A111661.
%F G.f.: Sum_{k>0} -(-1)^k * k^2 * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker( -3, k) * (x^k - x^(2*k)) / (1 + x^k)^3.
%e G.f. = q - 5*q^2 + 9*q^3 - 11*q^4 + 24*q^5 - 45*q^6 + 50*q^7 - 53*q^8 + 81*q^9 + ...
%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# #^2 JacobiSymbol[ -3, n/#] &]]; (* _Michael Somos_, Oct 06 2013 *)
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^5 QPochhammer[ q^3] QPochhammer[ q^6]^4 / QPochhammer[ q^2]^4, {q, 0, n}]; (* _Michael Somos_, Oct 06 2013 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, -(-1)^d * d^2 * kronecker( -3, n/d)))}; /* _Michael Somos_, Oct 06 2013 */
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^5 * eta(x^3 + A) * eta(x^6 + A)^4 / eta(x^2 + A)^4, n))};
%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor( n); prod( k=1, matsize(A)[1], if(p = A[k,1], e = A[k,2]; if( p==3, 9^e, if( p==2, -(4^(e+1) + 9*(-1)^(e+1)) / 5, if( p%6==1, ((p^2)^(e+1) - 1) / (p^2 - 1), ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1)))))))};
%Y Cf. A111661.
%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,mult
%O 1,2
%A _Michael Somos_, Jul 09 2012