A113416 Expansion of eta(q^2)^7*eta(q^4)/(eta(q)*eta(q^8)^2) in powers of q.

1, 2, -2, -4, -2, -8, 4, 16, -2, 14, 8, -20, 4, -24, -16, 16, -2, 36, -14, -36, 8, -32, 20, 48, 4, 42, 24, -40, -16, -56, -16, 64, -2, 40, -36, -64, -14, -72, 36, 48, 8, 84, 32, -84, 20, -56, -48, 96, 4, 114, -42, -72, 24, -104, 40, 80, -16, 72, 56, -116, -16, -120, -64, 112, -2, 96, -40, -132, -36, -96, 64, 144, -14

Offset

0,2

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Formula

Euler transform of period 8 sequence [2, -5, 2, -6, 2, -5, 2, -4, ...].

a(n) = 2*b(n) where b(n) is multiplicative and b(2^e) = -1 if e>0, b(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8).

G.f.: 1+2(Sum_{k>0} (2k-1)*(-1)^[k/2]*x^(2k-1)/(1+x^(2k-1))) = Product_{k>0} (1-x^(2k))^4*(1+x^k)^2/((1+x^(2k))*(1+x^(4k))^2).

Mathematica

f[p_, e_] := If[1 < Mod[p, 8] < 7, ((-p)^(e+1)-1)/(-p-1), (p^(e+1)-1)/(p-1)]; f[2, e_] := -1; a[0] = 1; a[1] = 2; a[n_] := 2 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Aug 22 2023 *)

Prog

(PARI) a(n)=if(n<=0, n==0, -2*sumdiv(n, d, d*(d%2)*(-1)^(n/d+(d+1)\4)))
(PARI) {a(n)=local(A, p, e); if(n<=0, n==0, A=factor(n); 2*prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, -1, p*=kronecker(2, p); (p^(e+1)-1)/(p-1)))))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^7*eta(x^4+A)/(eta(x+A)*eta(x^8+A))^2, n))}

Crossrefs

Sequence in context: A307313 A338042 A131999 * A303140 A370469 A103178

Adjacent sequences: A113413 A113414 A113415 * A113417 A113418 A113419

Keyword

sign

Author

Michael Somos, Oct 29 2005

Status

approved