%I #7 Nov 29 2022 04:04:28
%S 2,4,6,8,18,24,16,54,96,120,32,162,384,600,720,64,486,1536,3000,4320,
%T 5040,128,1458,6144,15000,25920,35280,40320,256,4374,24576,75000,
%U 155520,246960,322560,362880,512,13122,98304,375000,933120,1728720,2580480,3265920,3628800
%N Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns starting with fixed m, 2<k<=n, 1<=m<=k.
%H G. C. Greubel, <a href="/A104001/b104001.txt">Rows n = 3..50 of the triangle, flattened</a>
%H T. Mansour, <a href="http://arXiv.org/abs/math.CO/9911243">Permutations containing and avoiding certain patterns</a>
%F T(n, k) = (k-2)! * (k-1)^(n+1-k).
%F From _G. C. Greubel_, Nov 29 2022: (Start)
%F T(n, 3) = A000079(n-2).
%F T(n, 4) = 6*A000244(n-4).
%F T(n, 5) = 4!*A000302(n-5).
%F T(2*n-3, n) = A152684(n-1). (End)
%e Triangle begins as:
%e 2;
%e 4, 6;
%e 8, 18, 24;
%e 16, 54, 96, 120;
%e 32, 162, 384, 600, 720;
%e 64, 486, 1536, 3000, 4320, 5040;
%e 128, 1458, 6144, 15000, 25920, 35280, 40320;
%t Table[(k-1)!*(k-1)^(n-k), {n,3,15}, {k,3,n}]//Flatten (* _G. C. Greubel_, Nov 29 2022 *)
%o (Magma) [Factorial(k-1)*(k-1)^(n-k): k in [3..n], n in [3..15]]; // _G. C. Greubel_, Nov 29 2022
%o (SageMath)
%o def A104001(n,k): return factorial(k-1)*(k-1)^(n-k)
%o flatten([[A104001(n,k) for k in range(3,n+1)] for n in range(3,16)]) # _G. C. Greubel_, Nov 29 2022
%Y Cf. A000079, A000244, A000302, A137268,
%K nonn,tabl
%O 3,1
%A _Ralf Stephan_, Feb 26 2005