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A156184 A generalized recursion triangle sequence : m=1; t(n,k)=(k + m - 1)*t(n - 1, k, m) + (m*n - k + 1 - m)*t(n - 1, k - 1, m).

%I A156184 #8 Feb 17 2024 23:44:06
%S A156184 1,1,1,1,2,1,1,4,4,1,1,7,16,7,1,1,11,53,53,11,1,1,16,150,318,150,16,1,
%T A156184 1,22,380,1554,1554,380,22,1,1,29,892,6562,12432,6562,892,29,1,1,37,
%U A156184 1987,25038,82538,82538,25038,1987,37,1,1,46,4270,89023,480380,825380,480380
%N A156184 A generalized recursion triangle sequence : m=1; t(n,k)=(k + m - 1)*t(n - 1, k, m) + (m*n - k + 1 - m)*t(n - 1, k - 1, m).
%C A156184 Row sums are: A054091;
%C A156184 {1, 2, 4, 10, 32, 130, 652, 3914, 27400, 219202, 1972820, ...}.
%C A156184 The sequence comes from a generalization of the recurrence for A008517.
%H A156184 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Second-OrderEulerianTriangle.html">Second-Order Eulerian Triangle</a>.
%F A156184 t(n,k) = (k + m - 1)*t(n - 1, k, m) + (m*n - k + 1 - m)*t(n - 1, k - 1, m).
%e A156184 {1},
%e A156184 {1, 1},
%e A156184 {1, 2, 1},
%e A156184 {1, 4, 4, 1},
%e A156184 {1, 7, 16, 7, 1},
%e A156184 {1, 11, 53, 53, 11, 1},
%e A156184 {1, 16, 150, 318, 150, 16, 1},
%e A156184 {1, 22, 380, 1554, 1554, 380, 22, 1},
%e A156184 {1, 29, 892, 6562, 12432, 6562, 892, 29, 1},
%e A156184 {1, 37, 1987, 25038, 82538, 82538, 25038, 1987, 37, 1},
%e A156184 {1, 46, 4270, 89023, 480380, 825380, 480380, 89023, 4270, 46, 1}
%t A156184 m = 1; e[n_, 0, m_] := 1;
%t A156184 e[n_, k_, m_] := 0 /; k >= n;
%t A156184 e[n_, k_, 1] := 1 /; k >= n;
%t A156184 e[n_, k_, m_] := (k + m - 1)e[n - 1, k, m] + (m*n - k + 1 - m)e[n - 1, k - 1, m];
%t A156184 Table[Table[e[n, k, m], {k, 0, n}], {n, 0, 10}];
%t A156184 Flatten[%]
%Y A156184 Cf. A054091, A054090, A008517, A156141.
%K A156184 nonn,tabl,uned
%O A156184 0,5
%A A156184 _Roger L. Bagula_, Feb 05 2009