A355989 -id:A355989 - 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!.

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)).

A370890 A(n, k) = 2^n*Pochhammer(k/2, floor((n+1)/2)). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 4, 3, 1, 0, 12, 16, 6, 4, 1, 0, 60, 32, 30, 8, 5, 1, 0, 120, 192, 60, 48, 10, 6, 1, 0, 840, 384, 420, 96, 70, 12, 7, 1, 0, 1680, 3072, 840, 768, 140, 96, 14, 8, 1, 0, 15120, 6144, 7560, 1536, 1260, 192, 126, 16, 9, 1

Offset: 0

Peter Luschny, Mar 04 2024

			The array starts:
[0] 1,  1,   1,   1,   1,    1,    1,    1,    1,    1, ...
[1] 0,  1,   2,   3,   4,    5,    6,    7,    8,    9, ...
[2] 0,  2,   4,   6,   8,   10,   12,   14,   16,   18, ...
[3] 0,  6,  16,  30,  48,   70,   96,  126,  160,  198, ...
[4] 0, 12,  32,  60,  96,  140,  192,  252,  320,  396, ...
[5] 0, 60, 192, 420, 768, 1260, 1920, 2772, 3840, 5148, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,   1;
[2] 0,   1,   1;
[3] 0,   2,   2,  1;
[4] 0,   6,   4,  3,  1;
[5] 0,  12,  16,  6,  4,  1;
[6] 0,  60,  32, 30,  8,  5, 1;
[7] 0, 120, 192, 60, 48, 10, 6, 1;

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Rows: A000012, A001477, A005843, A054000, A134582.
Columns: A000007, A081125, A355989.
Cf. A370419.

A := (n, k) -> 2^n*pochhammer(k/2, iquo(n+1,2)):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k):
seq(seq(T(n, k), k = 0..n), n = 0..10);

A370890[n_, k_] := 2^n*Pochhammer[k/2, Floor[(n+1)/2]];
Table[A370890[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 06 2024 *)

# Note the use of different kinds of division.
def A(n, k): return 2**n * rising_factorial(k/2, (n+1)//2)
for n in range(0, 9): print([A(n, k) for k in range(0, 9)])

cp's OEIS Frontend

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

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Mathematica

PARI

PARI

Formula

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

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Mathematica

PARI

PARI

Formula

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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Author

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Programs

Mathematica

PARI

Formula

A370890 A(n, k) = 2^n*Pochhammer(k/2, floor((n+1)/2)). Square array read by ascending antidiagonals.

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Maple

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