OFFSET
0,4
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 19 2012
STATUS
approved