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A174159 Triangle read by rows. T(n, k) = 2 * Eulerian(n, k - 1) - binomial(n - 1, k - 1)* binomial(n, k - 1) / k.

%I A174159 #10 Jul 27 2022 16:35:43
%S A174159 1,1,1,1,5,1,1,16,16,1,1,42,112,42,1,1,99,554,554,99,1,1,219,2277,
%T A174159 4657,2277,219,1,1,466,8390,30748,30748,8390,466,1,1,968,28880,175292,
%U A174159 310616,175292,28880,968,1,1,1981,95140,907864,2615416,2615416,907864,95140
%N A174159 Triangle read by rows. T(n, k) = 2 * Eulerian(n, k - 1) - binomial(n - 1, k - 1)* binomial(n, k - 1) / k.
%e A174159 [ 1] 1;
%e A174159 [ 2] 1,    1;
%e A174159 [ 3] 1,    5,     1;
%e A174159 [ 4] 1,   16,    16,      1;
%e A174159 [ 5] 1,   42,   112,     42,       1;
%e A174159 [ 6] 1,   99,   554,    554,      99,       1;
%e A174159 [ 7] 1,  219,  2277,   4657,    2277,     219,      1;
%e A174159 [ 8] 1,  466,  8390,  30748,   30748,    8390,    466,     1;
%e A174159 [ 9] 1,  968, 28880, 175292,  310616,  175292,  28880,   968,    1;
%e A174159 [10] 1, 1981, 95140, 907864, 2615416, 2615416, 907864, 95140, 1981, 1;
%p A174159 # Works also if based on (0, 0).
%p A174159 T := (n,k) -> `if`(k = 0, k^n, 2*combinat:-eulerian1(n, k-1) - binomial(n-1, k-1)* binomial(n, k-1) / k):
%p A174159 for n from 1 to 6 do seq(T(n,k), k=1..n) od;  # _Peter Luschny_, Jul 27 2022
%t A174159 Needs["Combinatorica`"];
%t A174159 T[n_, m_] := 2*Eulerian[n, m - 1] - Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m;
%t A174159 Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten
%Y A174159 Cf. A001263, A008292, A356118 (row sums).
%K A174159 nonn,tabl
%O A174159 1,5
%A A174159 _Roger L. Bagula_, Mar 10 2010
%E A174159 Edited by _Peter Luschny_, Jul 27 2022