A299504 Triangle read by rows, T(n,k) = (k+1)^(n-k)*k! for 0 <= k <= n.

1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 8, 18, 24, 24, 1, 16, 54, 96, 120, 120, 1, 32, 162, 384, 600, 720, 720, 1, 64, 486, 1536, 3000, 4320, 5040, 5040, 1, 128, 1458, 6144, 15000, 25920, 35280, 40320, 40320, 1, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880

Offset

0,5

Comments

With offset 1, T(n,k) is the number of permutations of [n] in which all entries left of 1 (if any) are excedances and 1 is in position n+1-k. An excedance of a permutation p is an entry p(i) such that p(i)>i. For example, T(4,2) = 4 counts 2314, 2413, 3412, 4312. - David Callan, Dec 12 2021

Links

Andrew Howroyd, Rows n=0..50 of triangle, flattened

Ron Graham, Eulerian Adventures with Don, 2018.

Example

Triangle starts:
[0] 1
[1] 1,   1
[2] 1,   2,    2
[3] 1,   4,    6,     6
[4] 1,   8,   18,    24,    24
[5] 1,  16,   54,    96,   120,    120
[6] 1,  32,  162,   384,   600,    720,    720
[7] 1,  64,  486,  1536,  3000,   4320,   5040,   5040
[8] 1, 128, 1458,  6144, 15000,  25920,  35280,  40320,  40320
[9] 1, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880

Maple

T := (n, k) -> (k+1)^(n-k)*k!: seq(seq(T(n, k), k=0..n), n=0..9);

Mathematica

Table[(k+1)^(n-k)*k!, {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 08 2018 *)

Prog

(Python)
from sympy import factorial
def T(n, k): return (k+1)**(n-k)*factorial(k)
for n in range(21): print([T(n, k) for k in range(n+1)]) # Indranil Ghosh, Mar 02 2018
(PARI) T(n, k) = {(k+1)^(n-k)*k!}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 08 2018
(Magma) [(k+1)^(n-k)*Factorial(k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 08 2018

Crossrefs

Sequence in context: A071946 A053495 A096747 * A300979 A371889 A167622

Adjacent sequences: A299501 A299502 A299503 * A299505 A299506 A299507

Keyword

nonn,tabl

Author

Peter Luschny, Mar 01 2018

Status

approved