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Triangle read by rows. T(n, k) = A356265(n, k) + A357078(n, k) for 0 <= k <= n.
1

%I #10 Sep 15 2022 14:44:27

%S 1,0,1,0,1,1,0,3,2,1,0,9,12,2,1,0,49,37,31,2,1,0,329,149,176,63,2,1,0,

%T 2561,794,853,702,127,2,1,0,22369,5599,3836,5709,2549,255,2,1,0,

%U 216225,47275,18422,37609,33949,8886,511,2,1,2291457,451176,107535,218506,344670,184653,29777,1023,2,1

%N Triangle read by rows. T(n, k) = A356265(n, k) + A357078(n, k) for 0 <= k <= n.

%C The triangle is the termwise sum of a refinement of the number of irreducible permutations A357078, and A356265, which is a refinement of the number of reducible permutations.

%e Triangle T(n, k) starts: [Row sums]

%e [0] 1; [1]

%e [1] 0, 1; [1]

%e [2] 0, 1, 1; [2]

%e [3] 0, 3, 2, 1; [6]

%e [4] 0, 9, 12, 2, 1; [24]

%e [5] 0, 49, 37, 31, 2, 1; [120]

%e [6] 0, 329, 149, 176, 63, 2, 1; [720]

%e [7] 0, 2561, 794, 853, 702, 127, 2, 1; [5040]

%e [8] 0, 22369, 5599, 3836, 5709, 2549, 255, 2, 1; [40320]

%e [9] 0, 216225, 47275, 18422, 37609, 33949, 8886, 511, 2, 1; [362880]

%o (SageMath)

%o def A357079_triangle(dim):

%o T = A357078_triangle(dim)

%o return [[0^n] + [T[n][k+1] + A356265_row(n)[k]

%o for k in range(n)] for n in range(dim)]

%o A357079_triangle(9)

%Y Cf. A356265, A357078, A059438, A003319.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Sep 11 2022