A097109 - OEIS

%I #27 Jan 22 2024 01:25:04

%S 1,1,0,-2,-3,0,0,2,0,-2,0,0,6,2,0,0,-3,0,0,2,0,-4,0,0,0,1,0,-2,-6,0,0,

%T 2,0,0,0,0,6,2,0,-4,0,0,0,2,0,0,0,0,6,3,0,0,-6,0,0,0,0,-4,0,0,0,2,0,

%U -4,-3,0,0,2,0,0,0,0,0,2,0,-2,-6,0,0,2,0,-2,0,0,12,0,0,0,0,0,0,4,0,-4,0,0,0,2,0,0,-3,0,0,2,0

%N G.f.: s(2)^2*s(3)^3/(s(1)*s(6)^2), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

%C Coefficients are multiplicative [Fine].

%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.36).

%H G. C. Greubel, <a href="/A097109/b097109.txt">Table of n, a(n) for n = 0..1000</a>

%F Fine gives an explicit formula for a(n) in terms of the divisors of n.

%F From _Michael Somos_, Sep 15 2006: (Start)

%F Expansion of (a(q) - 3*a(q^3) - 4*a(q^4) + 12*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.

%F Euler transform of period 6 sequence [ 1, -1, -2, -1, 1, -2, ...].

%F a(n) is multiplicative with a(2^e) = -3(1+(-1)^e)/2 if e>0, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6). (End)

%F a(3*n + 2) = 0. a(3*n) = A115978(n). a(3*n + 1) = A122861(n).

%F Sum_{k=0..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - _Amiram Eldar_, Jan 22 2024

%t QP = QPochhammer; s = QP[q^2]^2*(QP[q^3]^3/(QP[q]*QP[q^6]^2)) + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^2), n))} /* _Michael Somos_, Sep 15 2006 */

%o (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if( p==3, -2, if( p%6==1, e+1, 1-e%2))))))} /* _Michael Somos_, Sep 15 2006 */

%Y Cf. A010815, A115978, A122861, A248897.

%K sign,mult

%O 0,4

%A _N. J. A. Sloane_, Sep 16 2004