A362685 Triangle of numbers read by rows, T(n, k) = (n*(n-1))*Stirling2(k, 2), for n >= 1 and 1 <= k <= n.

0, 0, 2, 0, 6, 18, 0, 12, 36, 84, 0, 20, 60, 140, 300, 0, 30, 90, 210, 450, 930, 0, 42, 126, 294, 630, 1302, 2646, 0, 56, 168, 392, 840, 1736, 3528, 7112, 0, 72, 216, 504, 1080, 2232, 4536, 9144, 18360, 0, 90, 270, 630, 1350, 2790, 5670, 11430, 22950, 45990

Offset: 1

Author

Igor Victorovich Statsenko, May 01 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 2 nonempty boxes.

Examples

			n\k   1      2       3      4      5      6      7
1:    0
2:    0      2
3:    0      6      18
4:    0     12      36     84
5:    0     20      60    140    300
6:    0     30      90    210    450    930
7:    0     42     126    294    630   1302   2646
  ...
T(4,2) = 12: {1}{2}{}{} (12 ways).
T(4,3) = 36: {12}{3}{}{} (36 ways).
T(4,4) = 84: {123}{4}{}{} (84 ways).
		

Links

• Igor Victorovich 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), A068605 (right diagonal).

Programs

Maple

L := 2: 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 A362685(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(a-1)*stirling(n-comb(a,2),2) # Chai Wah Wu, Jun 20 2025

Formula

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