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

1, 1, 1, 1, 1, 4, 11, 4, 1, 1, 11, 72, 114, 72, 11, 1, 1, 26, 367, 1492, 2438, 1492, 367, 26, 1, 1, 57, 1630, 13992, 48965, 73120, 48965, 13992, 1630, 57, 1, 1, 120, 6680, 109538, 727982, 2169674, 3107640, 2169674, 727982, 109538, 6680, 120, 1

Offset: 1

Author

N. J. A. Sloane, Mar 16 2016

Keywords

• nonn
• tabf

Comments

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

Examples

			Triangle begins:
  1;
  1, 1, 1;
  1, 4, 11, 4, 1;
  1, 11, 72, 114, 72, 11, 1;
  1, 26, 367, 1492, 2438, 1492, 367, 26, 1;
  1, 57, 1630, 13992, 48965, 73120, 48965, 13992, 1630, 57, 1;
  ...
The matrices for row n=3, k=0..2 are:
  [2 0]  [1 1]  [0 2]
  [0 2]  [1 1]  [2 0]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1600 (first 40 rows)
• Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
• E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
• Andrew Howroyd, PARI Program

Crossrefs

Row sums are A000681.

Columns k=0..4 are A000012, A000295, A260585, A260575, A260582.

Central coefficients are A332729.

Cf. A008292, A269743, A269744.

Programs

PARI

\\ See link. - Andrew Howroyd, Feb 22 2020

Extensions

Terms a(26) and beyond from Andrew Howroyd, Feb 22 2020