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A368584 Table read by rows: T(n, k) = A124320(n + 1, k) * A048993(n, k).

%I A368584 #14 Jul 30 2025 11:26:48
%S A368584 1,0,2,0,3,12,0,4,60,120,0,5,210,1260,1680,0,6,630,8400,30240,30240,0,
%T A368584 7,1736,45360,327600,831600,665280,0,8,4536,216720,2772000,13305600,
%U A368584 25945920,17297280,0,9,11430,956340,20207880,162162000,575134560,908107200,518918400
%N A368584 Table read by rows: T(n, k) = A124320(n + 1, k) * A048993(n, k).
%H A368584 Elena L. Wang and Guoce Xin, <a href="https://arxiv.org/abs/2507.15654">On Ward Numbers and Increasing Schröder Trees</a>, arXiv:2507.15654 [math.CO], 2025. See p. 12.
%e A368584 Triangle starts:
%e A368584   [0] [1]
%e A368584   [1] [0, 2]
%e A368584   [2] [0, 3,   12]
%e A368584   [3] [0, 4,   60,    120]
%e A368584   [4] [0, 5,  210,   1260,    1680]
%e A368584   [5] [0, 6,  630,   8400,   30240,    30240]
%e A368584   [6] [0, 7, 1736,  45360,  327600,   831600,   665280]
%e A368584   [7] [0, 8, 4536, 216720, 2772000, 13305600, 25945920, 17297280]
%o A368584 (SageMath)
%o A368584 def Trow(n): return [rising_factorial(n+1, k)*stirling_number2(n, k) for k in range(n+1)]
%o A368584 for n in range(7): print(Trow(n))
%Y A368584 Cf. A124320 (rising factorial), A048993(Stirling2), A053492 (row sums), A213236 (alternating row sums), A001813 (main diagonal), A368583.
%K A368584 nonn,tabl
%O A368584 0,3
%A A368584 _Peter Luschny_, Jan 10 2024