A309047 Expansion of Product_{k>=0} (1 + x^(2^k) - x^(3*2^k)).

1, 1, 1, 0, 1, 0, 0, -1, 1, 1, 0, -1, 0, 0, -1, -1, 1, 2, 1, 0, 0, -1, -1, -1, 0, 1, 0, 0, -1, -1, -1, 0, 1, 2, 2, 1, 1, -1, 0, -1, 0, 0, -1, -1, -1, 0, -1, 0, 0, 1, 1, 1, 0, -1, 0, 0, -1, -1, -1, 0, -1, 0, 0, 1, 1, 1, 2, 1, 2, 0, 1, -1, 1, 0, -1, -2, 0, 1, -1, -1, 0, 1, 0, 0, -1, -1, -1, 0, -1, 0, 0, 1, -1, -1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, -1, -1, -1, 0, 1

Offset: 0

Author

Ilya Gutkovskiy, Jul 09 2019

Keywords

• sign

Crossrefs

Cf. A002487, A005590, A120562.

Programs

Mathematica

nmax = 109; CoefficientList[Series[Product[(1 + x^(2^k) - x^(3 2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 109; A[] = 1; Do[A[x] = (1 + x - x^3) A[x^2] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], a[(n - 1)/2] - a[(n - 3)/2]]; Table[a[n], {n, 0, 109}]

Formula

G.f. A(x) satisfies: A(x) = (1 + x - x^3) * A(x^2).

a(0) = a(1) = 1; a(2*n) = a(n), a(2*n+1) = a(n) - a(n-1).