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Triangle with T(n,k)=n!*(k-1)^k/k! where 1<=k<=n.
2

%I #16 Jul 15 2023 10:35:34

%S 0,0,1,0,3,8,0,12,32,81,0,60,160,405,1024,0,360,960,2430,6144,15625,0,

%T 2520,6720,17010,43008,109375,279936,0,20160,53760,136080,344064,

%U 875000,2239488,5764801,0,181440,483840,1224720,3096576,7875000,20155392,51883209,134217728

%N Triangle with T(n,k)=n!*(k-1)^k/k! where 1<=k<=n.

%H Seiichi Manyama, <a href="/A076482/b076482.txt">Rows n = 1..140, flattened</a>

%F T(n,k) = n*T(n, k-1) = A007778(k-1)*A008279(n,n-k) starting with T(n,n) = (n-1)^n = A007778(n-1).

%e Rows start

%e 0;

%e 0, 1;

%e 0, 3, 8;

%e 0, 12, 32, 81;

%e 0, 60, 160, 405, 1024;

%e 0, 360, 960, 2430, 6144, 15625;

%e ...

%t Table[n! (k-1)^k/k!,{n,0,10},{k,n}]//Flatten (* _Harvey P. Dale_, Nov 28 2019 *)

%Y Row sums are A076483.

%Y Cf. A001710, A007778, A008279.

%K nonn,tabl

%O 1,5

%A _Henry Bottomley_, Oct 14 2002