A355988 -id:A355988 - cp's OEIS frontend

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

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.

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

A374647 a(n) = n! / floor(2n/3)!.

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 30, 210, 336, 504, 5040, 7920, 11880, 154440, 240240, 360360, 5765760, 8910720, 13366080, 253955520, 390700800, 586051200, 12893126400, 19769460480, 29654190720, 741354768000, 1133836704000, 1700755056000, 47621141568000, 72684900288000

Offset: 0

Clark Kimberling, Jul 22 2024

nonn

Cf. A000142, A081125, A355988.

Mathematica
```
Table[n!/(Floor[2 n/3]!), {n, 0, 30}]
```

A374648 a(n) = n! / (floor(n/3))! - n! / (floor(n/2))!.

Original entry on oeis.org

0, 0, 0, 0, 12, 60, 240, 1680, 18480, 45360, 574560, 6320160, 19293120, 250810560, 3615131520, 10637827200, 173837664000, 2955240288000, 8874542476800, 168616307059200, 3378360124339200, 10123012406707200, 222987858828134400, 5128720753047091200

Offset: 0

Clark Kimberling, Jul 25 2024

nonn

Cf. A000142, A081125, A355988.

Table[n!/(Floor[n/3]!) - n!/(Floor[n/2]!), {n, 0, 24}]

a(n) = A355988(n) - A081125(n). - Vaclav Kotesovec, Jul 27 2024

cp's OEIS Frontend

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

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

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A374647 a(n) = n! / floor(2n/3)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A374648 a(n) = n! / (floor(n/3))! - n! / (floor(n/2))!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula