A269947 Triangle read by rows, Stirling cycle 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*T(n-1,k), for n>=0 and 0<=k<=n.

1, 0, 1, 0, 1, 1, 0, 8, 9, 1, 0, 216, 251, 36, 1, 0, 13824, 16280, 2555, 100, 1, 0, 1728000, 2048824, 335655, 15055, 225, 1, 0, 373248000, 444273984, 74550304, 3587535, 63655, 441, 1, 0, 128024064000, 152759224512, 26015028256, 1305074809, 25421200, 214918, 784, 1

Offset: 0

Author

Peter Luschny, Mar 22 2016

Keywords

• nonn
• tabl

Examples

			Triangle starts:
1,
0, 1,
0, 1,       1,
0, 8,       9,       1,
0, 216,     251,     36,     1,
0, 13824,   16280,   2555,   100,   1,
0, 1728000, 2048824, 335655, 15055, 225, 1.
		

Links

• Peter Luschny, The P-transform.

Crossrefs

Variant: A249677.

Cf. A007318 (order 0), A132393 (order 1), A269944 (order 2).

Cf. A000442, A000537, A255433.

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*T(n-1, k))) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

Mathematica

T[n_, k_] := T[n, k] = Which[n == k, 1, k < 0 || k > n, 0, True, T[n - 1, k - 1] + (n - 1)^3 T[n - 1, k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

Formula

T(n,1) = ((n-1)!)^3 for n>=1 (cf. A000442).

T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).

Row sums: Product_{k=1..n} ((k-1)^3+1) for n>=0 (cf. A255433).