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

1, -5, 9, -11, 24, -45, 50, -53, 81, -120, 120, -99, 170, -250, 216, -203, 288, -405, 362, -264, 450, -600, 528, -477, 601, -850, 729, -550, 840, -1080, 962, -821, 1080, -1440, 1200, -891, 1370, -1810, 1530, -1272, 1680, -2250, 1850, -1320, 1944, -2640, 2208

Offset: 1

Author

Michael Somos, Jul 09 2012

Keywords

• sign
• mult

Comments

Zagier (2009) writes "... associated to the weight 3 Eisenstein series g(z) = Sigma b(n)q^n = q - 5q^2 + 9q^3 - 11q^4 + ...".

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			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 + ...
		

References

• 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.

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..2000
• D. Zagier, Integral solutions of Apery-like recurrence equations.

Crossrefs

Cf. A111661.

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.)

Programs

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# #^2 JacobiSymbol[ -3, n/#] &]]; (* Michael Somos, Oct 06 2013 *)
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 *)

PARI

{a(n) = if( n<1, 0, sumdiv( n, d, -(-1)^d * d^2 * kronecker( -3, n/d)))}; /* Michael Somos, Oct 06 2013 */

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))};

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)))))))};

Formula

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.

Euler transform of period 6 sequence [ -5, -1, -6, -1, -5, -6, ...]. - Michael Somos, Oct 06 2013

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).

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.

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.