A362791 Triangle of numbers read by rows, T(n, k) = (n*(n-1)*(n-2))*Stirling2(k, 3), for n >= 1 and 1 <= k <= n.

0, 0, 0, 0, 0, 6, 0, 0, 24, 144, 0, 0, 60, 360, 1500, 0, 0, 120, 720, 3000, 10800, 0, 0, 210, 1260, 5250, 18900, 63210, 0, 0, 336, 2016, 8400, 30240, 101136, 324576, 0, 0, 504, 3024, 12600, 45360, 151704, 486864, 1524600, 0, 0, 720, 4320, 18000, 64800, 216720, 695520, 2178000, 6717600

Offset: 1

Author

Igor Victorovich Statsenko, May 04 2023

Keywords

• nonn
• tabl

Comments

T(n, k) is the number of ways to distribute k labeled items into n labeled boxes so that there are exactly 3 nonempty boxes.

Examples

			n\k   1      2      3      4      5      6      7
1:    0
2:    0      0
3:    0      0      6
4:    0      0     24    144
5:    0      0     60    360   1500
6:    0      0    120    720   3000  10800
7:    0      0    210   1260   5250  18900  63210
  ...
T(4,3) = 24:  {1}{2}{3}{} (24 ways).
T(4,4) = 144: {12}{3}{4}{} (144 ways).
		

Links

• I. V. Statsenko, Generalized layout problem, Innovation science No 4-2, State Ufa, Aeterna Publishing House, 2023, pp. 10-13. In Russian.

Crossrefs

Cf. A002024 (case L=1), A362685 (case L=2), A068605 (right diagonal).

Programs

Maple

L := 3: T := (n, k) -> pochhammer(-n, L)*Stirling2(k, L)*((-1)^L):
seq(seq(T(n, k), k = 1..n), n = 1..10);

Python

from math import isqrt, comb
from sympy.functions.combinatorial.numbers import stirling
def A362791(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(a-1)*(a-2)*stirling(n-comb(a,2),3) # Chai Wah Wu, Jun 20 2025

Formula

T(n, k) = (n!/(n - L)!) * Stirling2(k, L) with L = 3, T(1,1)=T(2,1)=T(2,2) = 0.