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A269742 Triangle of generalized Eulerian numbers T(n,k) = _2 read by rows, n >= 1, 0 <= k < 2*n.

%I A269742 #22 Feb 22 2024 20:18:31
%S A269742 1,1,1,1,1,4,11,4,1,1,11,72,114,72,11,1,1,26,367,1492,2438,1492,367,
%T A269742 26,1,1,57,1630,13992,48965,73120,48965,13992,1630,57,1,1,120,6680,
%U A269742 109538,727982,2169674,3107640,2169674,727982,109538,6680,120,1
%N A269742 Triangle of generalized Eulerian numbers T(n,k) = <n,k>_2 read by rows, n >= 1, 0 <= k < 2*n.
%C A269742 T(n,k) is the number of nonnegative integer n X n matrices with every row and column sum 2 and sum of entries below the main diagonal k. The case when every row and column sum is 1 is given by the Eulerian numbers (A008292). - _Andrew Howroyd_, Feb 22 2020
%H A269742 Andrew Howroyd, <a href="/A269742/b269742.txt">Table of n, a(n) for n = 1..1600</a> (first 40 rows)
%H A269742 Esther M. Banaian, <a href="http://digitalcommons.csbsju.edu/honors_thesis/24">Generalized Eulerian Numbers and Multiplex Juggling Sequences</a>, (2016). All College Thesis Program. Paper 24.
%H A269742 E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, <a href="http://arxiv.org/abs/1508.03673">A generalization of Eulerian numbers via rook placements</a>, arXiv:1508.03673 [math.CO], 2015.
%H A269742 Andrew Howroyd, <a href="/A269742/a269742.txt">PARI Program</a>
%e A269742 Triangle begins:
%e A269742   1;
%e A269742   1, 1, 1;
%e A269742   1, 4, 11, 4, 1;
%e A269742   1, 11, 72, 114, 72, 11, 1;
%e A269742   1, 26, 367, 1492, 2438, 1492, 367, 26, 1;
%e A269742   1, 57, 1630, 13992, 48965, 73120, 48965, 13992, 1630, 57, 1;
%e A269742   ...
%e A269742 The matrices for row n=3, k=0..2 are:
%e A269742   [2 0]  [1 1]  [0 2]
%e A269742   [0 2]  [1 1]  [2 0]
%o A269742 (PARI) \\ See link. - _Andrew Howroyd_, Feb 22 2020
%Y A269742 Row sums are A000681.
%Y A269742 Columns k=0..4 are A000012, A000295, A260585, A260575, A260582.
%Y A269742 Central coefficients are A332729.
%Y A269742 Cf. A008292, A269743, A269744.
%K A269742 nonn,tabf
%O A269742 1,6
%A A269742 _N. J. A. Sloane_, Mar 16 2016
%E A269742 Terms a(26) and beyond from _Andrew Howroyd_, Feb 22 2020