A216916 Triangle read by rows, T(n,k) for 0<=k<=n, generalizing A098742.

1, 1, 1, 3, 3, 1, 9, 12, 6, 1, 33, 51, 34, 10, 1, 135, 237, 193, 79, 15, 1, 609, 1188, 1132, 584, 160, 21, 1, 2985, 6381, 6920, 4268, 1510, 293, 28, 1, 15747, 36507, 44213, 31542, 13576, 3464, 497, 36, 1, 88761, 221400, 295314, 238261, 120206, 37839, 7231, 794

Offset: 0

Author

Peter Luschny, Sep 20 2012

Keywords

• nonn
• tabl

Comments

Full concordance with A098742 would require two zero rows at the top of the triangle which we omitted for simplicity.

Matrix inverse is A137338. - Peter Luschny, Sep 21 2012

Examples

			[0] [1]
[1] [1, 1]
[2] [3, 3, 1]
[3] [9, 12, 6, 1]
[4] [33, 51, 34, 10, 1]
[5] [135, 237, 193, 79, 15, 1]
[6] [609, 1188, 1132, 584, 160, 21, 1]
[7] [2985, 6381, 6920, 4268, 1510, 293, 28, 1]
[8] [15747, 36507, 44213, 31542, 13576, 3464, 497, 36, 1]
		

Programs

Sage

def A216916_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+1)*T[n-1,k]+(k+2)*T[n-1,k+1]
    return T
A216916_triangle(9)

Formula

Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1) + (k+1)*T(n-1,k) + (k+2)*T(n-1,k+1).