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Triangle of generalized Eulerian numbers T(n,k) = <n,k>_3 read by rows, n >= 1, 0 <= k <= 3*(n-1).
5

%I #17 Feb 22 2020 19:59:52

%S 1,1,1,1,1,1,4,11,23,11,4,1,1,11,72,325,595,595,325,72,11,1,1,26,367,

%T 3368,14679,34679,46800,34679,14679,3368,367,26,1,1,57,1630,28819,

%U 253247,1212440,3382133,5588593,5588593,3382133,1212440,253247,28819,1630,57,1

%N Triangle of generalized Eulerian numbers T(n,k) = <n,k>_3 read by rows, n >= 1, 0 <= k <= 3*(n-1).

%C T(n,k) is the number of nonnegative integer n X n matrices with every row and column sum 3 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 Andrew Howroyd, <a href="/A269743/b269743.txt">Table of n, a(n) for n = 1..925</a> (first 25 rows)

%H 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 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 Andrew Howroyd, <a href="/A269743/a269743.txt">PARI Program</a>

%e Triangle begins:

%e 1,

%e 1,1,1,1,

%e 1,4,11,23,11,4,1,

%e 1,11,72,325,595,595,325,72,11,1,

%e ...

%o (PARI) \\ See link. - _Andrew Howroyd_, Feb 22 2020

%Y Row sums are A001500.

%Y Columns k=0..4 are A000012, A000295, A260585, A260727, A260583.

%Y Cf. A008292, A269742, A269744.

%K nonn,tabf

%O 1,7

%A _N. J. A. Sloane_, Mar 16 2016

%E Terms a(23) and beyond from _Andrew Howroyd_, Feb 22 2020