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

0,9

Comments

Also called 3rd central factorial numbers.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 1..150, flattened.)

Aleks Žigon Tankosič, The Lah Numbers with Higher Level and the Lah Numbers of Order s, arXiv:2510.26842 [math.GM], 2025. See p. 4.

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.

Example

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.

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 *)

Crossrefs

Variant: A098436.

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

Cf. A000537, A023001, A098437.

Sequence in context: A388468 A388863 A388753 * A121935 A070060 A329085

Adjacent sequences: A269945 A269946 A269947 * A269949 A269950 A269951

Keyword

nonn,tabl

Author

Peter Luschny, Mar 22 2016

Status

approved