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

%I A299504 #24 Sep 08 2022 08:46:20
%S A299504 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,
%T A299504 720,720,1,64,486,1536,3000,4320,5040,5040,1,128,1458,6144,15000,
%U A299504 25920,35280,40320,40320,1,256,4374,24576,75000,155520,246960,322560,362880,362880
%N A299504 Triangle read by rows, T(n,k) = (k+1)^(n-k)*k! for 0 <= k <= n.
%C A299504 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
%H A299504 Andrew Howroyd, <a href="/A299504/b299504.txt">Rows n=0..50 of triangle, flattened</a>
%H A299504 Ron Graham, <a href="http://knuth80.elfbrink.se/wp-content/uploads/2018/01/Graham-Pitea-talk-5.pdf">Eulerian Adventures with Don</a>, 2018.
%e A299504 Triangle starts:
%e A299504 [0] 1
%e A299504 [1] 1,   1
%e A299504 [2] 1,   2,    2
%e A299504 [3] 1,   4,    6,     6
%e A299504 [4] 1,   8,   18,    24,    24
%e A299504 [5] 1,  16,   54,    96,   120,    120
%e A299504 [6] 1,  32,  162,   384,   600,    720,    720
%e A299504 [7] 1,  64,  486,  1536,  3000,   4320,   5040,   5040
%e A299504 [8] 1, 128, 1458,  6144, 15000,  25920,  35280,  40320,  40320
%e A299504 [9] 1, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880
%p A299504 T := (n,k) -> (k+1)^(n-k)*k!: seq(seq(T(n,k), k=0..n), n=0..9);
%t A299504 Table[(k+1)^(n-k)*k!, {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 08 2018 *)
%o A299504 (Python)
%o A299504 from sympy import factorial
%o A299504 def T(n, k): return (k+1)**(n-k)*factorial(k)
%o A299504 for n in range(21): print([T(n, k) for k in range(n+1)]) # _Indranil Ghosh_, Mar 02 2018
%o A299504 (PARI) T(n,k) = {(k+1)^(n-k)*k!}
%o A299504 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 08 2018
%o A299504 (Magma) [(k+1)^(n-k)*Factorial(k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 08 2018
%K A299504 nonn,tabl
%O A299504 0,5
%A A299504 _Peter Luschny_, Mar 01 2018