A338688 -id:A338688 - cp's OEIS frontend

A338683 a(n) = - Sum_{d|n} (-n/d)^d * binomial(d+n/d-1, d).

1, 3, 10, 3, 26, 13, 50, -177, 352, -84, 122, -996, 170, -153, 9704, -13313, 290, -6518, 362, -2771, 107986, 17073, 530, -805070, 394376, 99984, 1203580, 1196313, 842, -4500745, 962, -13313025, 14199222, 2316234, 33547310, -19898071, 1370, 10418613, 168405072

Offset: 1

Seiichi Manyama, Apr 23 2021

sign

Cf. A327238, A338662, A338682, A338684, A338688.

a[n_] := -DivisorSum[n, (-n/#)^# * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Apr 24 2021 *)

a(n) = -sumdiv(n, d, (-n/d)^d*binomial(d+n/d-1, d));

N=66; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+k*x^k)^k))

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

If p is prime, a(p) = (-1)^(p-1) + p^2.

A338689 a(n) = Sum_{d|n} (-1)^(d-1) * (n/d)^n * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 3, 28, 223, 3126, 44660, 823544, 16514047, 387538588, 9951176994, 285311670612, 8903202187413, 302875106592254, 11107259264162760, 437894348359764856, 18444492187995996159, 827240261886336764178, 39345059356329821149097

Offset: 1

Seiichi Manyama, Apr 24 2021

nonn

Cf. A217670, A338661, A338684, A338688.

a[n_] := DivisorSum[n, (-1)^(# - 1) * (n/#)^n * Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)

a(n) = sumdiv(n, d, (-1)^(d-1)*(n/d)^n*binomial(d+n/d-2, d-1));

N=20; x='x+O('x^N); Vec(sum(k=1, N, (k*x/(1+(k*x)^k))^k))

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

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

cp's OEIS Frontend

A338683 a(n) = - Sum_{d|n} (-n/d)^d * binomial(d+n/d-1, d).

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A338689 a(n) = Sum_{d|n} (-1)^(d-1) * (n/d)^n * binomial(d+n/d-2, d-1).

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula