A121860 -id:A121860 - cp's OEIS frontend

A354844 a(n) = n! * Sum_{d|n} (n/d)^d / (d! * (n/d)!).

1, 3, 4, 29, 6, 1027, 8, 26889, 272170, 861851, 12, 515592013, 14, 1530809295, 668366899216, 9382044672017, 18, 1405750464518419, 20, 1393382139935385621, 4274473667143680022, 30537988748467223, 24, 211745638285336995840025

Offset: 1

Seiichi Manyama, Jun 08 2022

nonn

Cf. A121860, A354843, A354845.

a[n_] := n! * DivisorSum[n, (n/#)^#/(#! * (n/#)!) &]; Array[a, 25] (* Amiram Eldar, Jun 08 2022 *)

a(n) = n!*sumdiv(n, d, (n/d)^d/(d!*(n/d)!));

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

E.g.f.: Sum_{k>0} (exp(k * x^k) - 1)/k!.

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

A354862 a(n) = n! * Sum_{d|n} (n/d)! / d!.

Original entry on oeis.org

1, 5, 37, 601, 14401, 520801, 25401601, 1626189601, 131682257281, 13168407228481, 1593350922240001, 229442707280223361, 38775788043632640001, 7600054676241325858561, 1710012252750418295078401, 437763137119219420513804801, 126513546505547170185216000001

Offset: 1

Seiichi Manyama, Jun 09 2022

nonn

Cf. A020549, A057625, A121860, A354843, A354863.

a[n_] := n! * DivisorSum[n, (n/#)! / #! &]; Array[a, 17] (* Amiram Eldar, Aug 30 2023 *)

PARI
```
a(n) = n!*sumdiv(n, d, (n/d)!/d!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k!*(exp(x^k)-1))))

from math import factorial
from sympy import divisors
def A354862(n):
    f = factorial(n)
    return sum(f*(a := factorial(n//d))//(b:= factorial(d)) + (f*b//a if d**2 < n else 0) for d in divisors(n,generator=True) if d**2 <= n) # Chai Wah Wu, Jun 09 2022

E.g.f.: Sum_{k>0} k! * (exp(x^k) - 1).

If p is prime, a(p) = 1 + (p!)^2 = A020549(p).

A354897 a(n) = n! * Sum_{d|n} d^n / (d! * (n/d)!).

Original entry on oeis.org

1, 5, 28, 353, 3126, 94237, 823544, 72042497, 585825130, 157671732881, 285311670612, 790577855833537, 302875106592254, 5876819345289651137, 55890419425648520176, 73205730667453550166017, 827240261886336764178, 1474631675630757976051079425

Offset: 1

Seiichi Manyama, Jun 11 2022

nonn

Cf. A121860, A354844, A354890, A354892, A354893, A354898, A354899.

a[n_] := n! * DivisorSum[n, #^n/(#! * (n/#)!) &]; Array[a, 18] (* Amiram Eldar, Jun 11 2022 *)

a(n) = n!*sumdiv(n, d, d^n/(d!*(n/d)!));

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp((k*x)^k)-1)/k!)))

E.g.f.: Sum_{k>0} (exp((k * x)^k) - 1)/k!.

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

A354898 a(n) = n! * Sum_{d|n} d^(n - d) / (d! * (n/d)!).

Original entry on oeis.org

1, 2, 2, 26, 2, 2582, 2, 268802, 7348322, 51120722, 2, 299332756802, 2, 7157951760962, 18701679546950402, 613777679843328002, 2, 3250742570192384467202, 2, 29411516073133093829529602, 1146522800008167069616128002, 4017001663590220290585602, 2

Offset: 1

Seiichi Manyama, Jun 11 2022

nonn

Robert Israel, Table of n, a(n) for n = 1..305

Cf. A121860, A354845, A354891, A354893, A354897.

f:= proc(n) local d; n! * add(d^(n-d)/(d! * (n/d)!), d = numtheory:-divisors(n)) end proc:
map(f, [$1..30]); # Robert Israel, Jul 10 2023

a[n_] := n! * DivisorSum[n, #^(n - #)/(#! * (n/#)!) &]; Array[a, 23] (* Amiram Eldar, Jun 11 2022 *)

a(n) = n!*sumdiv(n, d, d^(n-d)/(d!*(n/d)!));

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

E.g.f.: Sum_{k>0} (exp((k * x)^k) - 1)/(k^k * k!).

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

A354899 a(n) = n! * Sum_{d|n} d^d / (d! * (n/d)!).

Original entry on oeis.org

1, 5, 28, 281, 3126, 48517, 823544, 16995617, 387692650, 10047310481, 285311670612, 8932562801857, 302875106592254, 11119129387084097, 437899615088648176, 18451106376806703617, 827240261886336764178, 39349894934527426209025

Offset: 1

Seiichi Manyama, Jun 11 2022

nonn

Cf. A121860, A354844, A354888, A354897, A354900.

a[n_] := n! * DivisorSum[n, #^#/(#! * (n/#)!) &]; Array[a, 18] (* Amiram Eldar, Jun 11 2022 *)

a(n) = n!*sumdiv(n, d, d^d/(d!*(n/d)!));

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k^k*(exp(x^k)-1)/k!)))

E.g.f.: Sum_{k>0} k^k * (exp(x^k) - 1)/k!.

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

A386877 Triangle read by rows: T(n, k) = n! / (k! * (n/k)!) if k divides n otherwise 0; T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 6, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 60, 60, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 840, 0, 840, 0, 0, 0, 1, 0, 1, 0, 10080, 0, 0, 0, 0, 0, 1, 0, 1, 15120, 0, 0, 15120, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Peter Bala and Peter Luschny, Aug 09 2025

			Triangle starts:
  [ 0] [1]
  [ 1] [0, 1]
  [ 2] [0, 1,     1]
  [ 3] [0, 1,     0,     1]
  [ 4] [0, 1,     6,     0,   1]
  [ 5] [0, 1,     0,     0,   0,     1]
  [ 6] [0, 1,    60,    60,   0,     0, 1]
  [ 7] [0, 1,     0,     0,   0,     0, 0, 1]
  [ 8] [0, 1,   840,     0, 840,     0, 0, 0, 1]
  [ 9] [0, 1,     0, 10080,   0,     0, 0, 0, 0, 1]
  [10] [0, 1, 15120,     0,   0, 15120, 0, 0, 0, 0, 1]
  [11] [0, 1,     0,     0,   0,     0, 0, 0, 0, 0, 0, 1]

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Cf. A121860 (row sums), A113704 (sign).

A386877[n_, k_] := Which[k == 0, Boole[n == 0], Divisible[n, k], n!/(k!*(n/k)!), True, 0];
Table[A386877[n, k], {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Aug 09 2025 *)

F = factorial
def T(n, k):
    if k == 0: return 0**n
    return F(n)/(F(k)*F(n//k)) if k.divides(n) else 0
for n in range(33): print([T(n,k) for k in srange(n+1)])

sign(T(n, k)) = A113704(n, k).

A356004 a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d! * (k/d)!).

Original entry on oeis.org

1, 4, 14, 64, 322, 2054, 14380, 116722, 1060580, 10636042, 116996464, 1411275650, 18346583452, 256869465610, 3856674412952, 61743633813634, 1049641774831780, 18896533652098442, 359034139389870400, 7182372973523436802, 150833211474559084844

Offset: 1

Seiichi Manyama, Jul 22 2022

nonn

Cf. A002627, A121860, A355886, A355991.

a[n_] := n! * Sum[DivisorSum[k, 1/(#!*(k/#)!) &], {k, 1, n}]; Array[a, 21] (* Amiram Eldar, Jul 22 2022 *)

a(n) = n!*sum(k=1, n, sumdiv(k,d,1/(d!*(k/d)!)));

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

E.g.f.: (1/(1-x)) * Sum_{k>0} (exp(x^k) - 1)/k!.

a(n) = n! * Sum_{k=1..n} A121860(k)/k!.

A363737 a(n) = n! * Sum_{d|n} (-1)^(d+1) / (d! * (n/d)!).

Original entry on oeis.org

1, 0, 2, -6, 2, 0, 2, -1680, 10082, 0, 2, -665280, 2, 0, 3632428802, -36843206400, 2, 0, 2, -670442572800, 3379030566912002, 0, 2, -71812452903064473600, 1077167364120207360002, 0, 10002268381116211200002, -3497296636753920000, 2, 0, 2

Offset: 1

Seiichi Manyama, Jun 18 2023

sign

Cf. A121860, A132962, A363736.

a[n_] := n! * DivisorSum[n, (-1)^(#+1)/(#! * (n/#)!) &]; Array[a, 30] (* Amiram Eldar, Jul 03 2023 *)

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

E.g.f.: Sum_{k>0} (1 - exp(-x^k))/k!.

If p is an odd prime, a(p) = 2.

A370581 a(n) = n! * Sum_{d|n} d/(d! * (n/d)!).

Original entry on oeis.org

1, 3, 4, 17, 6, 307, 8, 5049, 30250, 105851, 12, 25945933, 14, 77837775, 14529715216, 147891744017, 18, 13435316294419, 20, 7606841430988821, 16895152834560022, 183030822374423, 24, 387276381308571955225, 5385836820601036800026, 485735643993600027

Offset: 1

Seiichi Manyama, Feb 23 2024

Cf. A121860, A370579, A370580.

PARI
```
a(n) = n!*sumdiv(n, d, d/(d!*(n/d)!));
```

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

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

E.g.f.: Sum_{k>0} x^k/k! * exp(x^k).

cp's OEIS Frontend

A354844 a(n) = n! * Sum_{d|n} (n/d)^d / (d! * (n/d)!).

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A354862 a(n) = n! * Sum_{d|n} (n/d)! / d!.

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A354897 a(n) = n! * Sum_{d|n} d^n / (d! * (n/d)!).

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A354898 a(n) = n! * Sum_{d|n} d^(n - d) / (d! * (n/d)!).

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A354899 a(n) = n! * Sum_{d|n} d^d / (d! * (n/d)!).

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A386877 Triangle read by rows: T(n, k) = n! / (k! * (n/k)!) if k divides n otherwise 0; T(n, 0) = 0^n.

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A356004 a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d! * (k/d)!).

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PARI

PARI

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A363737 a(n) = n! * Sum_{d|n} (-1)^(d+1) / (d! * (n/d)!).

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