A217537 Triangle read by rows, T(n,k) = T(n-1,k-1) + k*T(n-1,k) + (k+1)*T(n-1,k+1), T(0,0) = 1, n >= 0, k >= 0.

1, 0, 1, 1, 1, 1, 1, 4, 3, 1, 4, 11, 13, 6, 1, 11, 41, 55, 35, 10, 1, 41, 162, 256, 200, 80, 15, 1, 162, 715, 1274, 1176, 595, 161, 21, 1, 715, 3425, 6791, 7182, 4361, 1526, 294, 28, 1, 3425, 17722, 38553, 45781, 32256, 13755, 3486, 498, 36, 1, 17722, 98253

Offset: 1

Author

Peter Luschny, Oct 06 2012

Keywords

• nonn
• tabl

Comments

Related to set partitions without singletons, T(n,0) = A000296(n).

Examples

			[0]    1,
[1]    0,    1,
[2]    1,    1,    1,
[3]    1,    4,    3,    1,
[4]    4,   11,   13,    6,    1,
[5]   11,   41,   55,   35,   10,    1,
[6]   41,  162,  256,  200,   80,   15,    1,
[7]  162,  715, 1274, 1176,  595,  161,   21,    1,
[8]  715, 3425, 6791, 7182, 4361, 1526,  294,   28,    1
		

Links

• Peter Luschny, Aigner Triangles

Crossrefs

Row sums are A217924, A000296 (first column).

Programs

Mathematica

T[0, 0] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k*T[n - 1, k] + (k + 1)*T[n - 1, k + 1]; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

Sage

def A217537_triangle(dim):
    T = matrix(ZZ,dim,dim)
    for n in range(dim): T[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            T[n,k] = T[n-1,k-1]+k*T[n-1,k]+(k+1)*T[n-1,k+1]
    return T
A217537_triangle(9)

Formula

From Mélika Tebni, Mar 26 2022: (Start)

E.g.f. column k: exp(exp(x) - 1 - x)*(exp(x) - 1)^k / k!, k >= 0.

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n. (End)