A328774 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} phi(n) * x^n, where phi = A000010.

1, 1, 1, 1, 2, -2, 4, -3, 4, -7, 14, -21, 30, -38, 50, -79, 128, -190, 286, -419, 598, -895, 1386, -2121, 3178, -4733, 7122, -10796, 16414, -25011, 38056, -57722, 87568, -133308, 203618, -311318, 475536, -726069, 1109718, -1698185, 2601166, -3987305, 6114666, -9378656, 14389676

Offset: 1

Author

Ilya Gutkovskiy, Oct 27 2019

Keywords

• sign

Comments

Inverse weigh transform of A000010.

Crossrefs

Cf. A000010, A299069, A320778.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; numtheory[phi](n)-b(n, n-1) end:
seq(a(n), n=1..45);  # Alois P. Heinz, Oct 27 2019

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = EulerPhi[n] - b[n, n - 1]; Array[a, 45]