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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.

%I A362685 #64 Jun 21 2025 11:21:03
%S A362685 0,0,2,0,6,18,0,12,36,84,0,20,60,140,300,0,30,90,210,450,930,0,42,126,
%T A362685 294,630,1302,2646,0,56,168,392,840,1736,3528,7112,0,72,216,504,1080,
%U A362685 2232,4536,9144,18360,0,90,270,630,1350,2790,5670,11430,22950,45990
%N A362685 Triangle of numbers read by rows, T(n, k) = (n*(n-1))*Stirling2(k, 2), for n >= 1 and 1 <= k <= n.
%C A362685 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.
%H A362685 Igor Victorovich 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 A362685 T(n, k) = (n!/(n - L)!) * Stirling2(k, L) with L = 2, T(1, 1) = 0.
%e A362685 n\k   1      2       3      4      5      6      7
%e A362685 1:    0
%e A362685 2:    0      2
%e A362685 3:    0      6      18
%e A362685 4:    0     12      36     84
%e A362685 5:    0     20      60    140    300
%e A362685 6:    0     30      90    210    450    930
%e A362685 7:    0     42     126    294    630   1302   2646
%e A362685   ...
%e A362685 T(4,2) = 12: {1}{2}{}{} (12 ways).
%e A362685 T(4,3) = 36: {12}{3}{}{} (36 ways).
%e A362685 T(4,4) = 84: {123}{4}{}{} (84 ways).
%p A362685 L := 2: T := (n, k) -> pochhammer(-n, L)*Stirling2(k, L)*((-1)^L):
%p A362685 seq(seq(T(n, k), k = 1..n), n = 1..10);
%o A362685 (Python)
%o A362685 from math import isqrt, comb
%o A362685 from sympy.functions.combinatorial.numbers import stirling
%o A362685 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
%Y A362685 Cf. A002024 (case L=1), A068605 (right diagonal).
%K A362685 nonn,tabl
%O A362685 1,3
%A A362685 _Igor Victorovich Statsenko_, May 01 2023