A229143 - OEIS

%I #32 Sep 04 2023 02:03:02

%S 1,0,-3,1,0,0,2,0,0,0,0,-3,2,0,0,1,0,0,2,0,-6,0,0,0,1,0,0,2,0,0,2,0,0,

%T 0,0,0,2,0,-6,0,0,0,2,0,0,0,0,-3,3,0,0,2,0,0,0,0,-6,0,0,0,2,0,0,1,0,0,

%U 2,0,0,0,0,0,2,0,-3,2,0,0,2,0,0,0,0,-6

%N Expansion of (b(q^3) - b(q)) / 3 in powers of q where b() is a cubic AGM theta function.

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

%C Rogers and Zudilin (2011) page 6: "This identity can be verified by eliminating b(q) with b(q^{1/3}) - b(q) = 3c(q^3) - c(q)."

%C The zeros of the g.f. A(q) where q = exp(2 Pi i t) are of the form t = (m/2 + sqrt(-3)/18) / n where m is an odd integer and n is in A004611. For example, (1/2 + sqrt(-3)/18) / 1, (1/2 + sqrt(-3)/18) / 7, (5/2 + sqrt(-3)/18) / 13.

%H G. C. Greubel, <a href="/A229143/b229143.txt">Table of n, a(n) for n = 1..2500</a>

%H M. Rogers and W. Zudilin, <a href="http://arXiv.org/abs/1102.1153">On the Mahler measure of 1 + X + 1/X + Y + 1/Y</a>, arXiv:1102.1153 [math.NT], 2011.

%F Expansion of c(q^3) / 3 - c(q^9) in powers of q where c() is a cubic AGM theta function.

%F Expansion of (a(q) - 4*a(q^3) + 3*a(q^9)) / 6 in powers of q where a() is a cubic AGM theta function.

%F Expansion of (eta(q^3)^4 - eta(q)^3 * eta(q^9)) / (3 * eta(q^3) * eta(q^9)) in powers of q.

%F a(n) is multiplicative with a(3) = -3, a(3^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1 + (-1)^e) / 2 if p == 2 (mod 3).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

%F a(3*n + 2) = a(4*n + 2) = a(9*n) = a(9*n + 6) = 0. a(3*n + 1) = A033687(n). a(9*n + 3) = -3 * A033687(n).

%F From _Michael Somos_, Jun 16 2015: (Start)

%F a(4*n) = a(n). a(6*n + 1) = A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).

%F a(n) = Sum_{d|n} A259024(n/d) * [ 0, 1, 0, -2, 0, 1][mod(d, 6) + 1]. (End)

%e G.f. = q - 3*q^3 + q^4 + 2*q^7 - 3*q^12 + 2*q^13 + q^16 + 2*q^19 - 6*q^21 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3]^4 - QPochhammer[ q^9] QPochhammer[ q]^3) / (3 QPochhammer[ q^3] QPochhammer[ q^9]), {q, 0, n}]; (* _Michael Somos_, Jun 16 2015 *)

%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^9]^4 - 3 q^2 QPochhammer[ q^3] QPochhammer[ q^27]^3) / (QPochhammer[ q^3] QPochhammer[ q^9]), {q, 0, n}]; (* _Michael Somos_, Jun 16 2015 *)

%t f[p_, e_] := If[Mod[p, 3] == 1, e+1, (1 + (-1)^e) / 2]; f[3, 1] = -3; f[3, e_] := 0; a[n_] := Times @@ f @@@ FactorInteger[n]; a[0] = 0; a[1] = 1; Array[a, 100, 0] (* _Amiram Eldar_, Sep 04 2023 *)

%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3, -3 * (e==1), p%3==1, e+1, !(e%2))))};

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^9 + A)^4 - 3 * x^2 * eta(x^3 + A) * eta(x^27 + A)^3) / (eta(x^3 + A) * eta(x^9 + A)), n))};

%o (Magma) A := Basis( ModularForms( Gamma1(27), 1), 85); A[2] - 3*A[4] + A[5] + 2*A[8] - 3*A[13] + 2*A[14] + A[15]; /* _Michael Somos_, Jun 16 2015 */

%Y Cf. A004611, A033687, A097195, A121361, A123884, A259024.

%K sign,easy,mult

%O 1,3

%A _Michael Somos_, Sep 23 2013