A338682 a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-1, d).

1, 1, 4, 0, 6, 3, 8, -8, 20, 0, 12, -12, 14, -7, 72, -65, 18, 10, 20, -61, 142, -33, 24, -203, 152, -52, 248, -183, 30, 121, 32, -617, 398, -102, 828, -619, 38, -133, 600, -896, 42, 140, 44, -870, 2864, -207, 48, -4438, 1766, 751, 1192, -1587, 54, -348, 4424, -3011, 1598, -348, 60

Offset

1,3

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k >= 1} (1 - 1/(1 + x^k)^k).

G.f.: - Sum_{k >= 1} (-x)^k/(1 - x^k)^(k+1).

If p is prime, a(p) = (-1)^(p-1) + p.

Mathematica

a[n_] := DivisorSum[n, (-1)^(# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 60] (* Amiram Eldar, Apr 24 2021 *)

Prog

(PARI) a(n) = sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-1, d));
(PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+x^k)^k))
(PARI) N=66; x='x+O('x^N); Vec(-sum(k=1, N, (-x)^k/(1-x^k)^(k+1)))

Crossrefs

Cf. A081543, A217670, A318636, A338683, A338684.

Sequence in context: A360044 A127447 A226775 * A305731 A279433 A096272

Adjacent sequences: A338679 A338680 A338681 * A338683 A338684 A338685

Keyword

sign

Author

Seiichi Manyama, Apr 23 2021

Status

approved