A216154 Triangle read by rows, T(n,k) n>=0, k>=0, generalization of A000255.

1, 1, 1, 3, 4, 1, 11, 21, 9, 1, 53, 128, 78, 16, 1, 309, 905, 710, 210, 25, 1, 2119, 7284, 6975, 2680, 465, 36, 1, 16687, 65821, 74319, 35035, 7945, 903, 49, 1, 148329, 660064, 857836, 478464, 133630, 19936, 1596, 64, 1, 1468457, 7275537, 10690812, 6879684, 2279214, 419958, 44268, 2628, 81, 1

Offset: 0

Author

Peter Luschny, Sep 19 2012

Keywords

• nonn
• tabl

Examples

			     1,
     1,      1,
     3,      4,      1,
    11,     21,      9,      1,
    53,    128,     78,     16,      1,
   309,    905,    710,    210,     25,      1,
  2119,   7284,   6975,   2680,    465,     36,      1,
16687,  65821,  74319,  35035,   7945,    903,     49,      1,
148329, 660064, 857836, 478464, 133630,  19936,   1596,     64,      1,
		

Crossrefs

A000255 (col. 0), A110450 (diag. n,n-2).

Cf. A111596, A271703.

Programs

Maple

A216154 := proc(n,k) local L, Z;
L := (n,k) -> `if`(k<0 or k>n,0,(n-k)!*C(n,n-k)*C(n-1,n-k)):
Z := (n,k) -> add(C(-j,-n)*L(j,k), j=0..n);
Z(n+1, k+1) end:
seq(seq(A216154(n,k), k=0..n), n=0..9); # Peter Luschny, Apr 13 2016

Mathematica

T[0, 0] = 1; T[0, ] = 0; T[n, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (2k+1) T[n-1, k] + (k+1) (k+2) 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 A216154_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(1+2*k)*M[n-1,k]+(k+1)*(k+2)*M[n-1,k+1]
    return M
A216154_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)+(1+2*k)*T(n-1,k)+(k+1)*(k+2)*T(n-1,k+1).

Let Z(n, k) = Sum_{j=0..n} C(-j, -n)*L(j, k) where L denotes the unsigned Lah numbers A271703. Then T(n, k) = Z(n+1, k+1). - Peter Luschny, Apr 13 2016