A269946 Triangle read by rows, Lah numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^3+k^3)*T(n-1, k), for n>=0 and 0<=k<=n.

1, 0, 1, 0, 2, 1, 0, 18, 18, 1, 0, 504, 648, 72, 1, 0, 32760, 47160, 7200, 200, 1, 0, 4127760, 6305040, 1141560, 45000, 450, 1, 0, 895723920, 1416456720, 283704120, 13741560, 198450, 882, 1, 0, 308129028480, 498072032640, 106386981120, 5876519040, 106616160, 691488, 1568, 1

Offset: 0

Author

Peter Luschny, Mar 22 2016

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[1]
[0, 1]
[0, 2,       1]
[0, 18,      18,      1]
[0, 504,     648,     72,      1]
[0, 32760,   47160,   7200,    200,   1]
[0, 4127760, 6305040, 1141560, 45000, 450, 1]
		

Links

• Peter Luschny, The P-transform.

Crossrefs

Cf. A038207 (order 0), A111596 (order 1), A268434 (order 2).

Cf. A255433, A163102, A269947, A269948.

Programs

Maple

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + ((n-1)^3+k^3) * T(n-1, k) )) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

Mathematica

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^3 + k^3)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

Formula

T(n,k) = Sum_{j=k..n} A269947(n,j)*A269948(j,k).

T(n,1) = Product_{k=1..n} (k-1)^3+1 for n>=1 (cf. A255433).

T(n,n-1) = (n-1)^2*n^2/2 for n>=1 (cf. A163102).