A256550 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.

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, 0, 203, 1092, 1575, 875, 210, 21, 1, 0, 877, 5684, 10192, 7350, 2450, 392, 28, 1, 0, 4140, 31572, 68208, 61152, 26460, 5880, 672, 36, 1

Offset: 0

Author

Peter Luschny, Apr 01 2015

Keywords

• nonn
• tabl
• easy

Examples

			Triangle starts:
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;
0,  203, 1092, 1575,  875,  210,  21,  1;
		

Crossrefs

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

Programs

Sage

def T(dim) :
    M = matrix(ZZ, dim)
    for n in range(dim) :
        M[n, n] = 1
        for k in range(n) :
            M[n,k] = (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
    E = M.exp()/exp(1)
    for n in range(dim) :
        for k in range(n) :
            M[n,k] = E[n,k]/factorial(n-k+1)
    return M
T(8) # Computes the sequence as a lower triangular matrix.

Formula

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

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

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

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