A355991 -id:A355991 - cp's OEIS frontend

A355989 a(n) = n! / (2 * floor(n/2)!).

1, 3, 6, 30, 60, 420, 840, 7560, 15120, 166320, 332640, 4324320, 8648640, 129729600, 259459200, 4410806400, 8821612800, 167610643200, 335221286400, 7039647014400, 14079294028800, 323823762662400, 647647525324800, 16191188133120000, 32382376266240000

Offset: 2

Seiichi Manyama, Jul 22 2022

Column 2 of A355996.

Cf. A355990, A355991,

a[n_] := n!/(2 * Floor[n/2]!); Array[a, 25, 2] (* Amiram Eldar, Jul 22 2022 *)

PARI
```
a(n) = n!/(2*(n\2)!);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2)*(exp(x^2)-1)/(2*(1-x))))

from math import factorial, floor
def a(n): return int(factorial(n)/(2*factorial(floor(n/2))))
print([a(n) for n in range(2, 30)]) # Javier Rivera Romeu, Jul 22 2022

from sympy import rf
def A355989(n): return rf((m:=n+1>>1)+(n+1&1),m)>>1 # Chai Wah Wu, Jul 22 2022

E.g.f.: (1 - x^2) * (exp(x^2) - 1)/(2 * (1 - x)).
a(n) = A081125(n)/2.
From Amiram Eldar, Jul 26 2022: (Start)
Sum_{n>=2} 1/a(n) = 3*exp(1/4)*sqrt(Pi)*erf(1/2) - 2, where erf is the error function.
Sum_{n>=2} (-1)^n/a(n) = 2 - exp(1/4)*sqrt(Pi)*erf(1/2). (End)

A355990 a(n) = n! / (6 * floor(n/3)!).

Original entry on oeis.org

1, 4, 20, 60, 420, 3360, 10080, 100800, 1108800, 3326400, 43243200, 605404800, 1816214400, 29059430400, 494010316800, 1482030950400, 28158588057600, 563171761152000, 1689515283456000, 37169336236032000, 854894733428736000, 2564684200286208000

Offset: 3

Seiichi Manyama, Jul 22 2022

Column 3 of A355996.

Cf. A355988, A355989, A355991,

a[n_] := n!/(6 * Floor[n/3]!); Array[a, 22, 3] (* Amiram Eldar, Jul 22 2022 *)

PARI
```
a(n) = n!/(6*(n\3)!);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)*(exp(x^3)-1)/(6*(1-x))))

E.g.f.: (1 - x^3) * (exp(x^3) - 1)/(6 * (1 - x)).

a(n) = A355988(n)/6.

A355996 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) = n!/(k! * floor(n/k)!).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 30, 20, 5, 1, 1, 60, 60, 30, 6, 1, 1, 420, 420, 210, 42, 7, 1, 1, 840, 3360, 840, 336, 56, 8, 1, 1, 7560, 10080, 7560, 3024, 504, 72, 9, 1, 1, 15120, 100800, 75600, 15120, 5040, 720, 90, 10, 1, 1, 166320, 1108800, 831600, 166320, 55440, 7920, 990, 110, 11, 1

Offset: 1

Seiichi Manyama, Jul 22 2022

			Triangle begins:
  1;
  1,   1;
  1,   3,    1;
  1,   6,    4,   1;
  1,  30,   20,   5,   1;
  1,  60,   60,  30,   6,  1;
  1, 420,  420, 210,  42,  7, 1;
  1, 840, 3360, 840, 336, 56, 8, 1;
  ...

Row sums give A355991.

Column k=1..3 give A000012, A355989, A355990.

T[n_, k_] := n!/(k!*Floor[n/k]!); Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 22 2022 *)

PARI
```
T(n, k) = n!/(k!*(n\k)!);
```

E.g.f. of column k: (1 - x^k) * (exp(x^k) - 1)/(k! * (1 - x)).

A355987 a(n) = n! * Sum_{k=1..n} 1/floor(n/k)!.

Original entry on oeis.org

1, 3, 13, 61, 421, 2641, 23521, 203281, 2071441, 22407841, 286403041, 3453468481, 51122111041, 759194916481, 12216117513601, 203300293996801, 3811792426041601, 69634723878720001, 1444704854104512001, 29725332567567436801, 658231789483184716801

Offset: 1

Seiichi Manyama, Jul 22 2022

nonn

Cf. A007489, A081125, A345683, A355886, A355988, A355991.

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

PARI
```
a(n) = n!*sum(k=1, n, 1/(n\k)!);
```

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

E.g.f.: (1/(1-x)) * Sum_{k>0} (1 - x^k) * (exp(x^k) - 1).
a(n) ~ c * n! * n, where c = 0.59962032... - Vaclav Kotesovec, Aug 03 2022
Conjecture: c = Sum_{k>=1} 1/((k+1)!*k) = 2 - exp(1) - A001620 + A091725. - Vaclav Kotesovec, Sep 24 2023

A356011 a(n) = n! * Sum_{k=1..n} 1/(k! * floor(n/k)).

Original entry on oeis.org

1, 2, 6, 17, 80, 337, 2240, 14681, 117010, 1023941, 10900472, 108881665, 1375544846, 17732140805, 247041590476, 3605768497217, 59990390084690, 977383707751621, 18214603019184800, 337615168055209601, 6763842079452393622, 141262515443311046885

Offset: 1

Seiichi Manyama, Jul 23 2022

nonn

Row sums of A356013.

Cf. A345682, A345683, A355991, A356015.

Table[n! * Sum[1/(k!*Floor[n/k]), {k,1,n}], {n,1,25}] (* Vaclav Kotesovec, Aug 11 2025 *)

PARI
```
a(n) = n!*sum(k=1, n, 1/(k!*(n\k)));
```

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

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

a(n) ~ exp(1) * (n-1)!. - Vaclav Kotesovec, Aug 11 2025

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!.

cp's OEIS Frontend

A355989 a(n) = n! / (2 * floor(n/2)!).

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A355990 a(n) = n! / (6 * floor(n/3)!).

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A355996 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) = n!/(k! * floor(n/k)!).

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Formula

A355987 a(n) = n! * Sum_{k=1..n} 1/floor(n/k)!.

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Formula

A356011 a(n) = n! * Sum_{k=1..n} 1/(k! * floor(n/k)).

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Formula

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

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