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A214262 Expansion of eta(q)^5 * eta(q^3) * eta(q^6)^4 / eta(q^2)^4 in powers of q.

%I A214262 #23 Sep 06 2018 02:41:43
%S A214262 1,-5,9,-11,24,-45,50,-53,81,-120,120,-99,170,-250,216,-203,288,-405,
%T A214262 362,-264,450,-600,528,-477,601,-850,729,-550,840,-1080,962,-821,1080,
%U A214262 -1440,1200,-891,1370,-1810,1530,-1272,1680,-2250,1850,-1320,1944,-2640,2208
%N A214262 Expansion of eta(q)^5 * eta(q^3) * eta(q^6)^4 / eta(q^2)^4 in powers of q.
%C A214262 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 A214262 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D A214262 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 A214262 Indranil Ghosh, <a href="/A214262/b214262.txt">Table of n, a(n) for n = 1..2000</a>
%H A214262 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 A214262 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 A214262 Euler transform of period 6 sequence [ -5, -1, -6, -1, -5, -6, ...]. - _Michael Somos_, Oct 06 2013
%F A214262 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 A214262 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 A214262 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 A214262 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 A214262 a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# #^2 JacobiSymbol[ -3, n/#] &]]; (* _Michael Somos_, Oct 06 2013 *)
%t A214262 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 A214262 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, -(-1)^d * d^2 * kronecker( -3, n/d)))}; /* _Michael Somos_, Oct 06 2013 */
%o A214262 (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 A214262 (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 A214262 Cf. A111661.
%Y A214262 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 A214262 sign,mult
%O A214262 1,2
%A A214262 _Michael Somos_, Jul 09 2012