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A362685
Triangle of numbers read by rows, T(n, k) = (n*(n-1))*Stirling2(k, 2), for n >= 1 and 1 <= k <= n.
1
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
1,3
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.
LINKS
Igor Victorovich 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 = 2, T(1, 1) = 0.
EXAMPLE
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).
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);
CROSSREFS
Cf. A002024 (case L=1), A068605 (right diagonal).
Sequence in context: A307581 A261566 A247443 * A171734 A278746 A280217
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved