A269948 Triangle read by rows, Stirling set 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)+k^3*T(n-1, k), for n>=0 and 0<=k<=n.

1, 0, 1, 0, 1, 1, 0, 1, 9, 1, 0, 1, 73, 36, 1, 0, 1, 585, 1045, 100, 1, 0, 1, 4681, 28800, 7445, 225, 1, 0, 1, 37449, 782281, 505280, 35570, 441, 1, 0, 1, 299593, 21159036, 33120201, 4951530, 130826, 784, 1

Offset: 0

Author

Peter Luschny, Mar 22 2016

Keywords

• nonn
• tabl

Comments

Also called 3rd central factorial numbers.

Examples

			1,
0, 1,
0, 1, 1,
0, 1, 9,     1,
0, 1, 73,    36,     1,
0, 1, 585,   1045,   100,    1,
0, 1, 4681,  28800,  7445,   225,   1,
0, 1, 37449, 782281, 505280, 35570, 441, 1.
		

Crossrefs

Variant: A098436.

Cf. A007318 (order 0), A048993 (order 1), A269945 (order 2).

Cf. A000537, A023001, A098437.

Programs

Maple

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + k^3*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n,k), k=0..n) od;

Mathematica

T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k^3*T[n - 1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 25 2019 *)

Formula

T(n,2) = (8^(n-1)-1)/7 for n>=1 (cf. A023001).

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

Row sums: A098437.