A353160 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} prime(n+1) * x^n.

3, 2, 0, 4, -4, 6, -8, 11, 0, -18, 40, -67, 88, -78, -4, 205, -524, 926, -1234, 1060, 140, -2998, 7900, -14132, 18858, -16280, -2710, 48783, -127826, 228784, -306494, 263582, 55144, -829538, 2160120, -3864533, 5172792, -4406277, -1118324, 14531101, -37606792

Offset: 1

Author

Ilya Gutkovskiy, Apr 28 2022

Keywords

• sign

Comments

Inverse weigh transform of odd primes.

Crossrefs

Cf. A006005, A065091, A305871, A352609, A353161.

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; ithprime(n+1)-b(n, n-1) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 28 2022

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] = Prime[n + 1] - b[n, n - 1]; Table[a[n], {n, 1, 41}]