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

0,4

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

Links

Table of n, a(n) for n=0..52.

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).

Example

[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]

Prog

(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)

Crossrefs

Sequence in context: A122919 A188513 A260301 * A157401 A143911 A185422

Adjacent sequences: A216913 A216914 A216915 * A216917 A216918 A216919

Keyword

nonn,tabl

Author

Peter Luschny, Sep 20 2012

Status

approved