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Triangle: m=3; e(n,k,n) = (k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m); t(n,k) = e(n,k,m) + e(n,n-k,m).
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%I #10 Feb 18 2024 05:15:35

%S 2,1,1,1,6,1,1,30,30,1,1,159,360,159,1,1,1119,3639,3639,1119,1,1,

%T 10932,41262,57414,41262,10932,1,1,136764,582642,898632,898632,582642,

%U 136764,1,1,2031933,9957168,16634718,17182152,16634718,9957168,2031933,1,1

%N Triangle: m=3; e(n,k,n) = (k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m); t(n,k) = e(n,k,m) + e(n,n-k,m).

%F m=3; e(n,k,n) = (k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m);

%F t(n,k) = e(n,k,m) + e(n,n-k,m).

%e {2},

%e {1, 1},

%e {1, 6, 1},

%e {1, 30, 30, 1},

%e {1, 159, 360, 159, 1},

%e {1, 1119, 3639, 3639, 1119, 1},

%e {1, 10932, 41262, 57414, 41262, 10932, 1},

%e {1, 136764, 582642, 898632, 898632, 582642, 136764, 1},

%e {1, 2031933, 9957168, 16634718, 17182152, 16634718, 9957168, 2031933, 1},...

%t m = 3; e[n_, 0, m_] := 1;

%t e[n_, k_, m_] := 0 /; k >= n;

%t e[n_, k_, 1] := 1 /; k >= n;

%t e[n_, k_, m_] := (k + m - 1)e[n - 1, k, m] + (m*n - k + 1 - m)e[n - 1, k - 1, m];

%t Table[Table[e[n, k, m], {k, 0, n - 1}], {n, 1, 10}];

%t Table[Table[e[n, k, m] + e[n, n - k, m], {k, 0, n}], {n, 0, 10}];

%t Flatten[%]

%Y Cf. A054091, A054090, A008517, A156141.

%K nonn,tabl,less,uned

%O 0,1

%A _Roger L. Bagula_, Feb 05 2009