A300278 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + n*x^n).

1, 1, 4, 5, 14, 19, 42, 57, 115, 170, 287, 433, 694, 1061, 1709, 2461, 3740, 5635, 8243, 12256, 18255, 26135, 37826, 54209, 78315, 110488, 159418, 224514, 315414, 442790, 618665, 855640, 1199409, 1642334, 2288904, 3144738, 4303994, 5862294, 8031872, 10869290, 14749050

Offset: 1

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

• nonn

Comments

Moebius transform of A022629.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000837, A022629, A078374, A085410, A300274, A300275, A300276, A300277.

Programs

Mathematica

nn = 41; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + n x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 41}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A022629(d).