A112848 - OEIS

%I #22 Jan 28 2024 04:47:36

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

%T 2,-1,0,0,0,-2,2,-2,-4,0,0,4,2,0,0,0,0,-2,3,-1,0,2,0,2,0,-2,-4,0,0,0,

%U 2,-2,-4,1,0,0,2,0,0,0,0,2,2,-2,-2,2,0,4,2,0,-2,0,0,-4,0,-2,0,0,0,0,4,0,-4,0,0,2,2,-3,0,1,0,0,2,-2,0

%N Expansion of eta(q)*eta(q^2)*eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

%H G. C. Greubel, <a href="/A112848/b112848.txt">Table of n, a(n) for n = 1..1000</a>

%F Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -2, ...].

%F Moebius transform is period 18 sequence [1, -2, -3, 2, -1, 6, 1, -2, 0, 2, -1, -6, 1, -2, 3, 2, -1, 0, ...].

%F Multiplicative with a(2^e) = (-1)^e, 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).

%F G.f.: Sum_{k>0} Kronecker(-3, k) x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} (1-x^k)(1-x^(2k))(1+x^(9k))(1+x^(6k)+x^(12k)).

%F a(3n) = -2*A092829(n). a(3n+1) = A093829(3n+1) = A033687(n). a(3n+2) = A093829(3n+2). a(6n)/2 = A093829(n). a(6n+1) = A097195(n). a(6n+3) = -2*A033762(n). a(6n+5) = 0.

%F Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - _Amiram Eldar_, Jan 23 2024

%t QP = QPochhammer; s = QP[q]*QP[q^2]*(QP[q^18]^2/(QP[q^6]*QP[q^9])) + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)

%t f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (-1)^e; f[3, e_]:= -2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 28 2024 *)

%o (PARI) {a(n)=if(n<1, 0, if(n%3==0, n/=3; -2,1)* sumdiv(n,d,kronecker(-12,d) -if(d%2==0, 2*kronecker(-3,d/2))))}

%o (PARI) {a(n)=local(A); if (n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^18+A)^2/ eta(x^6+A)/eta(x^9+A), n))}

%Y Cf. A033687, A033762, A092829, A093829, A097195, A248897, A255648 (absolute values).

%K sign,mult

%O 1,3

%A _Michael Somos_, Sep 22 2005