A080964 Euler transform of period-16 sequence [2,-3,2,1,2,-3,2,-6,2,-3,2,1,2,-3,2,-3,...].

1, 2, 0, 0, 4, 4, 0, 0, 2, -2, 0, 0, -8, -4, 0, 0, -4, 0, 0, 0, 8, -8, 0, 0, -8, -2, 0, 0, -16, 4, 0, 0, 6, -8, 0, 0, 12, 4, 0, 0, 8, 8, 0, 0, -8, 4, 0, 0, -8, 2, 0, 0, 24, -4, 0, 0, 0, 8, 0, 0, -16, 4, 0, 0, 12, 8, 0, 0, 16, 0, 0, 0, 10, -8, 0, 0, -24, 0, 0, 0, -8, -6, 0, 0, 16, 8, 0, 0, -24, -8, 0, 0, -16, -8, 0, 0, 8, 0, 0

Offset: 0

Author

Michael Somos, Feb 28 2003

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000 (1501 terms from G. C. Greubel)

Programs

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[eta[q^2]^5 *eta[q^8]^7/(eta[q]^2*eta[q^4]^4*eta[q^16]^3), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 70}] (* G. C. Greubel, Jul 02 2018 *)

PARI

a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^-2*eta(X^2)^5*eta(X^4)^-4*eta(X^8)^7*eta(X^16)^-3,n))

Formula

a(4*n+2) = a(4*n+3) = 0.

a(n) = 2*A072071(n) - A072070(n).

a(4*n) = A080965(n).

a(4*n+1) = 2*A080966(n).

Expansion of eta(q^2)^5*eta(q^8)^7/(eta(q)^2*eta(q^4)^4*eta(q^16)^3) in powers of q. - G. C. Greubel, Jul 02 2018