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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.

%I A362791 #28 Jun 21 2025 11:22:06
%S A362791 0,0,0,0,0,6,0,0,24,144,0,0,60,360,1500,0,0,120,720,3000,10800,0,0,
%T A362791 210,1260,5250,18900,63210,0,0,336,2016,8400,30240,101136,324576,0,0,
%U A362791 504,3024,12600,45360,151704,486864,1524600,0,0,720,4320,18000,64800,216720,695520,2178000,6717600
%N 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.
%C A362791 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.
%H A362791 I. V. Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2023-04-2.pdf#page=10">Generalized layout problem</a>, Innovation science No 4-2, State Ufa, Aeterna Publishing House, 2023, pp. 10-13. In Russian.
%F A362791 T(n, k) = (n!/(n - L)!) * Stirling2(k, L) with L = 3, T(1,1)=T(2,1)=T(2,2) = 0.
%e A362791 n\k   1      2      3      4      5      6      7
%e A362791 1:    0
%e A362791 2:    0      0
%e A362791 3:    0      0      6
%e A362791 4:    0      0     24    144
%e A362791 5:    0      0     60    360   1500
%e A362791 6:    0      0    120    720   3000  10800
%e A362791 7:    0      0    210   1260   5250  18900  63210
%e A362791   ...
%e A362791 T(4,3) = 24:  {1}{2}{3}{} (24 ways).
%e A362791 T(4,4) = 144: {12}{3}{4}{} (144 ways).
%p A362791 L := 3: T := (n, k) -> pochhammer(-n, L)*Stirling2(k, L)*((-1)^L):
%p A362791 seq(seq(T(n, k), k = 1..n), n = 1..10);
%o A362791 (Python)
%o A362791 from math import isqrt, comb
%o A362791 from sympy.functions.combinatorial.numbers import stirling
%o A362791 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
%Y A362791 Cf. A002024 (case L=1),  A362685 (case L=2), A068605 (right diagonal).
%K A362791 nonn,tabl
%O A362791 1,6
%A A362791 _Igor Victorovich Statsenko_, May 04 2023