A269742 Triangle of generalized Eulerian numbers T(n,k) = <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

1,6

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

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

Example

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]

Prog

(PARI) \\ See link. - Andrew Howroyd, Feb 22 2020

Crossrefs

Row sums are A000681.

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

Central coefficients are A332729.

Cf. A008292, A269743, A269744.

Sequence in context: A115643 A091387 A091388 * A151873 A354274 A205136

Adjacent sequences: A269739 A269740 A269741 * A269743 A269744 A269745

Keyword

nonn,tabf

Author

N. J. A. Sloane, Mar 16 2016

Extensions

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

Status

approved