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

0,4

Links

Table of n, a(n) for n=0..54.

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

Example

     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,

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 *)

Prog

(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)

Crossrefs

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

Cf. A111596, A271703.

Sequence in context: A172094 A114608 A154602 * A325174 A109956 A123319

Adjacent sequences: A216151 A216152 A216153 * A216155 A216156 A216157

Keyword

nonn,tabl

Author

Peter Luschny, Sep 19 2012

Status

approved