A081543 -id:A081543 - cp's OEIS frontend

A357041 a(n) = Sum_{d|n} 2^(d-1) * binomial(d+n/d-1,d).

1, 4, 7, 18, 21, 66, 71, 196, 305, 648, 1035, 2526, 4109, 8774, 16875, 34288, 65553, 134860, 262163, 531506, 1051237, 2109594, 4194327, 8425348, 16779257, 33611984, 67123631, 134350206, 268435485, 537178750, 1073741855, 2148064768, 4295048345, 8591114580

Offset: 1

Seiichi Manyama, Feb 26 2023

nonn

Cf. A081543, A338682.

Cf. A360797.

a[n_] := DivisorSum[n, 2^(#-1) * Binomial[# + n/# - 1, #] &]; Array[a, 50] (* Amiram Eldar, Jul 31 2023 *)

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (1/(1-2*x^k)^k-1))/2)

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (2*x)^k/(1-x^k)^(k+1))/2)

from math import comb
from sympy import divisors
def A357041(n): return sum(comb(d+n//d-1,d)<Chai Wah Wu, Feb 27 2023

G.f.: (1/2) * Sum_{k>0} (1/(1 - 2 * x^k)^k - 1).
G.f.: (1/2) * Sum_{k>0} (2 * x)^k/(1 - x^k)^(k+1).
If p is prime, a(p) = p + 2^(p-1).

A358411 a(n) = Sum_{d|n} (d + n/d - 1)!/(d - 1)!.

Original entry on oeis.org

1, 4, 9, 34, 125, 762, 5047, 40468, 362949, 3629560, 39916811, 479007174, 6227020813, 87178331590, 1307674370745, 20922790251808, 355687428096017, 6402373709377404, 121645100408832019, 2432902008216565330, 51090942171709621965, 1124000727778086681754

Offset: 1

Seiichi Manyama, Nov 14 2022

nonn

Cf. A081543, A358389, A358410.

a[n_] := DivisorSum[n, (# + n/# - 1)!/(# - 1)! &]; Array[a, 22] (* Amiram Eldar, Aug 30 2023 *)

a(n) = sumdiv(n, d, (d+n/d-1)!/(d-1)!);

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*x^k/(1-x^k)^(k+1)))

G.f.: Sum_{k>0} k! * x^k/(1 - x^k)^(k+1).

If p is prime, a(p) = p + p!.

A366974 Expansion of Sum_{k >=1} x^(2*k)/(1-x^k)^(k+1).

Original entry on oeis.org

0, 1, 2, 4, 4, 9, 6, 14, 12, 20, 10, 42, 12, 35, 40, 59, 16, 96, 18, 121, 84, 77, 22, 281, 80, 104, 156, 281, 28, 521, 30, 407, 264, 170, 406, 1083, 36, 209, 416, 1418, 40, 1514, 42, 1068, 1632, 299, 46, 3532, 840, 1923, 884, 1847, 52, 3824, 2420, 5377, 1216, 464, 58

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Cf. A081543, A366975.

Cf. A318636.

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

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

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

A366975 Expansion of Sum_{k >=1} x^(3*k)/(1-x^k)^(k+1).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 5, 9, 8, 14, 9, 25, 11, 27, 24, 40, 15, 65, 17, 75, 55, 65, 21, 176, 44, 90, 110, 182, 27, 324, 29, 270, 197, 152, 195, 695, 35, 189, 324, 847, 39, 925, 41, 759, 1016, 275, 45, 2215, 377, 1182, 730, 1365, 51, 2338, 1418, 3072, 1025, 434, 57, 7536, 59

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Cf. A081543, A366974.

Cf. A318636.

a(n) = if(n<2, 0, sumdiv(n, d, binomial(d+n/d-3, d)));

a(n) = Sum_{d|n} binomial(d+n/d-3,d) for n > 1.

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

cp's OEIS Frontend

A357041 a(n) = Sum_{d|n} 2^(d-1) * binomial(d+n/d-1,d).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Python

Formula

A358411 a(n) = Sum_{d|n} (d + n/d - 1)!/(d - 1)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A366974 Expansion of Sum_{k >=1} x^(2*k)/(1-x^k)^(k+1).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A366975 Expansion of Sum_{k >=1} x^(3*k)/(1-x^k)^(k+1).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula