A248686 Triangular array of multinomial coefficients: T(n,k) = n!/(n(1)!*n(2)!* ... *n(k)!), where n(i) = floor((n + i - 1)/k) for i = 1 .. k.

1, 1, 2, 1, 3, 6, 1, 6, 12, 24, 1, 10, 30, 60, 120, 1, 20, 90, 180, 360, 720, 1, 35, 210, 630, 1260, 2520, 5040, 1, 70, 560, 2520, 5040, 10080, 20160, 40320, 1, 126, 1680, 7560, 22680, 45360, 90720, 181440, 362880, 1, 252, 4200, 25200, 113400, 226800, 453600, 907200, 1814400, 3628800

Offset

1,3

Comments

T(n,k) is the number of permutations p of [n] such that p(i)<p(i+k) for i in [n-k]. T(4,2) = 6: 1234, 1243, 1324, 2134, 2143, 3142. - Alois P. Heinz, Feb 09 2023

Links

Clark Kimberling, Table of n, a(n) for n = 1..5000

Example

First seven rows:
  1
  1    2
  1    3     6
  1    6    12   24
  1   10    30   60    120
  1   20    90  180    360    720
  1   35   210  630   1260   2520   5040
  ...
Writing floor as [ ], the numbers comprising row 4 are
T(4,1) = 4!/[4/1]! = 24/24 = 1
T(4,2) = 4!/([4/2]![5/2]!) = 24/(2*2) = 6
T(4,3) = 4!/([4/3]![5/3]![6/3]! = 24/(1*1*2) = 12
T(4,4) = 4!/([4/4]![5/4]![6/4]![7/4]!) = 24/(1*1*1*1) = 24.

Maple

T:= (n, k)-> combinat[multinomial](n, floor((n+i)/k)$i=0..k-1):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Feb 09 2023

Mathematica

f[n_, k_] := f[n, k] = n!/Product[Floor[(n + i)/k]!, {i, 0, k - 1}]
t = Table[f[n, k], {n, 0, 10}, {k, 1, n}];
u = Flatten[t]  (* A248686 sequence *)
TableForm[t]    (* A248686 array *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 22}] (* A248687 *)

Crossrefs

Main diagonal is A000142.

T(2n,n) gives A000680.

Row sums give A248687.

Cf. A333706.

Sequence in context: A337407 A156914 A289656 * A059434 A292222 A182928

Adjacent sequences: A248683 A248684 A248685 * A248687 A248688 A248689

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Oct 11 2014

Status

approved