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A356265 Triangle read by rows. The reduced triangle of the partition triangle of reducible permutations (A356264). T(n, k) for n >= 1 and 0 <= k < n.

%I A356265 #11 Sep 11 2022 01:53:16
%S A356265 0,1,0,1,2,0,1,8,2,0,1,21,25,2,0,1,49,152,55,2,0,1,106,697,670,117,2,
%T A356265 0,1,223,2756,5493,2509,243,2,0,1,459,9966,36105,33669,8838,497,2,0,1,
%U A356265 936,34095,206698,342710,184305,29721,1007,2,0
%N A356265 Triangle read by rows. The reduced triangle of the partition triangle of reducible permutations (A356264). T(n, k) for n >= 1 and 0 <= k < n.
%H A356265 Peter Luschny, <a href="https://github.com/PeterLuschny/PermutationsWithLehmer/blob/main/PermutationsWithLehmer.ipynb">Permutations with Lehmer</a>, a SageMath Jupyter Notebook.
%e A356265 Triangle T(n, k) starts:                        [Row sums]
%e A356265 [1] [0]                                            [0]
%e A356265 [2] [1,   0]                                       [1]
%e A356265 [3] [1,   2,    0]                                 [3]
%e A356265 [4] [1,   8,    2,     0]                          [11]
%e A356265 [5] [1,  21,   25,     2,     0]                   [49]
%e A356265 [6] [1,  49,  152,    55,     2,    0]             [259]
%e A356265 [7] [1, 106,  697,   670,   117,    2,   0]        [1593]
%e A356265 [8] [1, 223, 2756,  5493,  2509,  243,   2, 0]     [11227]
%e A356265 [9] [1, 459, 9966, 36105, 33669, 8838, 497, 2, 0]  [89537]
%o A356265 (SageMath) # uses function A356264_row
%o A356265 @cache
%o A356265 def Pn(n: int, k: int) -> int:
%o A356265     if k == 0: return 0
%o A356265     if n == 0 or k == 1: return 1
%o A356265     return Pn(n, k - 1) + Pn(n - k, k) if k <= n else Pn(n, k - 1)
%o A356265 def reduce_parts(fun, n: int) -> list[int]:
%o A356265     funn: list[int] = fun(n)
%o A356265     return [sum(funn[Pn(n, k):Pn(n, k + 1)]) for k in range(n)]
%o A356265 def reduce_partition_triangle(fun, n: int) -> list[list[int]]:
%o A356265     return [reduce_parts(fun, k) for k in range(1, n)]
%o A356265 def A356265_row(n: int) -> list[int]:
%o A356265     return reduce_partition_triangle(A356264_row, n+1)[n-1]
%o A356265 for n in range(1, 8):
%o A356265     print(A356265_row(n))
%Y A356265 Cf. A356264 (partitions), A356291 (row sums).
%K A356265 nonn,tabl
%O A356265 1,5
%A A356265 _Peter Luschny_, Aug 16 2022