%I #8 Apr 19 2015 08:49:49

%S 1,0,1,0,1,1,0,2,3,1,0,5,12,6,1,0,15,50,40,10,1,0,52,225,250,100,15,1,

%T 0,203,1092,1575,875,210,21,1,0,877,5684,10192,7350,2450,392,28,1,0,

%U 4140,31572,68208,61152,26460,5880,672,36,1

%N Triangle read by rows, T(n,k) = EL(n,k)/(n-k+1)! and EL(n,k) the matrix-exponential of the unsigned Lah numbers scaled by exp(-1), for n>=0 and 0<=k<=n.

%F T(n+1,1) = Bell(n) = A000110(n).

%F T(n+2,2) = C(n+2,2)*Bell(n) = A105479(n+2).

%F T(n+1,n) = A000217(n).

%F T(n+2,n) = A008911(n+1).

%e Triangle starts:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 2, 3, 1;

%e 0, 5, 12, 6, 1;

%e 0, 15, 50, 40, 10, 1;

%e 0, 52, 225, 250, 100, 15, 1;

%e 0, 203, 1092, 1575, 875, 210, 21, 1;

%o (Sage)

%o def T(dim) :

%o M = matrix(ZZ, dim)

%o for n in range(dim) :

%o M[n, n] = 1

%o for k in range(n) :

%o M[n,k] = (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))

%o E = M.exp()/exp(1)

%o for n in range(dim) :

%o for k in range(n) :

%o M[n,k] = E[n,k]/factorial(n-k+1)

%o return M

%o T(8) # Computes the sequence as a lower triangular matrix.

%Y Cf. A000110, A000217, A008911, A105479, A256551 (matrix inverse).

%K nonn,tabl,easy

%O 0,8

%A _Peter Luschny_, Apr 01 2015