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A362789 Triangle read by rows. T(n, k) = FallingFactorial(n - k, k) * Stirling2(n - k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.

%I A362789 #7 May 04 2023 08:57:01
%S A362789 1,0,0,1,0,2,0,3,2,0,4,18,0,5,84,6,0,6,300,144,0,7,930,1500,24,0,8,
%T A362789 2646,10800,1200,0,9,7112,63210,23400,120,0,10,18360,324576,294000,
%U A362789 10800,0,11,45990,1524600,2857680,352800,720,0,12,112530,6717600,23496480,7056000,105840
%N A362789 Triangle read by rows. T(n, k) = FallingFactorial(n - k, k) * Stirling2(n - k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.
%e A362789 Triangle T(n, k) starts:
%e A362789 [0] 1;
%e A362789 [1] 0;
%e A362789 [2] 0, 1;
%e A362789 [3] 0, 2;
%e A362789 [4] 0, 3,    2;
%e A362789 [5] 0, 4,   18;
%e A362789 [6] 0, 5,   84,     6;
%e A362789 [7] 0, 6,  300,   144;
%e A362789 [8] 0, 7,  930,  1500,   24;
%e A362789 [9] 0, 8, 2646, 10800, 1200;
%p A362789 fallingFactorial := (x, n) -> (-1)^n * pochhammer(-x, n):
%p A362789 T := (n, k) -> fallingFactorial(n - k, k) * Stirling2(n - k, k):
%p A362789 seq(seq(T(n, k), k = 0..iquo(n,2)), n = 0..12);
%o A362789 (SageMath)
%o A362789 def A362789(n, k):
%o A362789     return falling_factorial(n - k, k) * stirling_number2(n - k, k)
%o A362789 for n in range(10):
%o A362789     print([A362789(n, k) for k in range(n//2 + 1)])
%Y A362789 Cf. A362790 (row sums), A362788, A362769.
%K A362789 nonn,tabf
%O A362789 0,6
%A A362789 _Peter Luschny_, May 04 2023