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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, 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
1,6
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.
LINKS
I. V. Statsenko, Generalized layout problem, Innovation science No 4-2, State Ufa, Aeterna Publishing House, 2023, pp. 10-13. In Russian.
FORMULA
T(n, k) = (n!/(n - L)!) * Stirling2(k, L) with L = 3, T(1,1)=T(2,1)=T(2,2) = 0.
EXAMPLE
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).
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);
CROSSREFS
Cf. A002024 (case L=1), A362685 (case L=2), A068605 (right diagonal).
Sequence in context: A002044 A230968 A028700 * A278014 A368846 A230337
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved