A355990 -id:A355990 - cp's OEIS frontend

A355991 a(n) = n! * Sum_{k=1..n} 1/(k! * floor(n/k)!).

1, 2, 5, 12, 57, 158, 1101, 5442, 28811, 212502, 2337513, 9422306, 122489967, 1654319046, 13917499277, 111631450818, 1897734663891, 23705612782022, 450406642858401, 3091477152208002, 51404897928720023, 1130752882197523686, 26007316290543044757

Offset: 1

Seiichi Manyama, Jul 22 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..464

Row sums of A355996.

Cf. A121860, A355886, A355987, A355989, A355990.

a[n_] := n! * Sum[1/(k! * Floor[n/k]!), {k, 1, n}]; Array[a, 23] (* Amiram Eldar, Jul 22 2022 *)

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)*(exp(x^k)-1)/k!)/(1-x)))

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

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

Original entry on oeis.org

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)

A355988 a(n) = n! / floor(n/3)!.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 360, 2520, 20160, 60480, 604800, 6652800, 19958400, 259459200, 3632428800, 10897286400, 174356582400, 2964061900800, 8892185702400, 168951528345600, 3379030566912000, 10137091700736000, 223016017416192000, 5129368400572416000

Offset: 0

Seiichi Manyama, Jul 22 2022

Cf. A081125, A355990.

a[n_] := n!/Floor[n/3]!; Array[a, 24, 0] (* Amiram Eldar, Jul 22 2022 *)

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

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

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

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)).

A356012 a(n) = n! / (6 * floor(n/3)).

Original entry on oeis.org

1, 4, 20, 60, 420, 3360, 20160, 201600, 2217600, 19958400, 259459200, 3632428800, 43589145600, 697426329600, 11856247603200, 177843714048000, 3379030566912000, 67580611338240000, 1216451004088320000, 26761922089943040000, 615524208068689920000

Offset: 3

Seiichi Manyama, Jul 23 2022

Column 3 of A356013.

Cf. A355990.

Table[n!/(6 Floor[n/3]),{n,3,30}] (* Harvey P. Dale, Jul 25 2024 *)

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

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

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

cp's OEIS Frontend

A355991 a(n) = n! * Sum_{k=1..n} 1/(k! * floor(n/k)!).

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

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A355988 a(n) = n! / 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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A356012 a(n) = n! / (6 * floor(n/3)).

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