%I #20 Jan 22 2024 01:18:36

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

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

%U 0,0,2,0,0,0,0,0,2,-4,1,0,0,-4,2,0,1,0,0,0,0,-4,0,0,0,0,4,0,2,0,0,0,2,-6,0,0,0,0,2,0,0

%N Expansion of eta(q)^2 * eta(q^6)^4 * eta(q^8) * eta(q^24) / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

%C Expansion of (a(q) - 2 * a(q^2) - a(q^4) + 2*a(q^8)) / 6 in powers of q where a() is a cubic AGM function.

%H G. C. Greubel, <a href="/A123484/b123484.txt">Table of n, a(n) for n = 1..1000</a>

%F Euler transform of period 24 sequence [ -2, 0, 0, 0, -2, -2, -2, -1, 0, 0, -2, 0, -2, 0, 0, -1, -2, -2, -2, 0, 0, 0, -2, -2, ...].

%F Moebius transform is period 24 sequence [ 1, -3, 0, 2, -1, 0, 1, 0, 0, 3, -1, 0, 1, -3, 0, 0, -1, 0, 1, -2, 0, 3, -1, 0, ...].

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A136748.

%F G.f.: x * Product_{k>0} (1 -x^(6*k)) * (1 - x^k + x^(2*k))^2 * (1 - x^(8*k)) * (1 + x^(12*k)) / (1 + x^(6*k)).

%F a(4*n) = a(6*n + 4) = a(6*n + 5) = 0. a(3*n) = a(n).

%F Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Pi/(2*sqrt(3)) = 0.906899... (A093766). - _Amiram Eldar_, Jan 22 2024

%e q - 2*q^2 + q^3 - 2*q^6 + 2*q^7 + q^9 + 2*q^13 - 4*q^14 - 2*q^18 + ...

%t QP = QPochhammer; s = QP[x]^2*QP[x^6]^4*QP[x^8]*(QP[x^24]/(QP[x^2]*QP[x^3]* QP[x^12])^2) + O[x]^105; CoefficientList[s, x] (* _Jean-François Alcover_, Nov 06 2015, adapted from PARI, updated Dec 06 2015 *)

%o (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, 1, d/2%2*-2)*kronecker(-12, n/d)))}

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

%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))}

%Y A033762(n) = a(2*n+1). A112604(n) = a(4*n+1). -2 * A033762(n) = a(4*n+2). A112605(n) = a(4*n+3). A097195(n) = a(6*n+1). A112606(n) = a(8*n+1). -2 * A112604(n) = a(8*n+2). A112608(n) = a(8*n+3). 2 * A112607(n) = a(8*n+5). -2 * A112605(n) = a(8*n+6). 2 * A112609(n) = a(8*n+7).

%Y A123884(n) = a(12*n+1). 2 * A121361(n) = a(12*n+7). A131961(n) = a(24*n+1). 2 * A131962(n) = a(24*n+7). A112608(n) = a(24*n+9). 2 * A131963(n) = a(24*n+13). 2 * A131964(n) = a(24*n+19).

%Y Cf. A093766, A136748.

%K sign,mult

%O 1,2

%A _Michael Somos_, Sep 28 2006, Apr 04 2008