A354043 Table read by rows: T(n, k) = (-1)^(n-k)*F(n, k)/k!, where F are the Faulhaber numbers A354042.

1, 0, 1, 0, 1, 1, 0, 4, 4, 1, 0, 36, 36, 10, 1, 0, 600, 600, 170, 20, 1, 0, 16584, 16584, 4720, 574, 35, 1, 0, 705600, 705600, 201040, 24640, 1568, 56, 1, 0, 43751232, 43751232, 12468960, 1531152, 98448, 3696, 84, 1, 0, 3790108800, 3790108800, 1080240480, 132713280, 8554896, 325152, 7812, 120, 1

Offset: 0

Author

Peter Luschny, May 17 2022

Keywords

• nonn
• tabl

Comments

I. Gessel and X. Viennot give two combinatorial interpretations for the Faulhaber numbers (see link). We quote their theorems 32 an 33, using our notation:

Theorem: T(n, k) is the number of row-strict tableaux of shape (n - k + 2, n - k + 1, ..., 2) - (n - k - 1, n - k - 2, ..., 0) with positive integer entries in which the largest entry in row i is at most n + 2 - i.

Theorem: T(n, k) is the number of sequences a_{1} a_{2} ... a_{3n-3k} of positive integers satisfying a_{3i-2} < a_{3i-1} < a_{3i}, a_{3i-1} >= a_{3i+1}, a_{3i} >= a_{3i+2}, and a_{3i} <= k + i + 1 for all i.

Examples

			Table starts:
  [0] 1;
  [1] 0,        1;
  [2] 0,        1,        1;
  [3] 0,        4,        4,        1;
  [4] 0,       36,       36,       10,       1;
  [5] 0,      600,      600,      170,      20,     1;
  [6] 0,    16584,    16584,     4720,     574,    35,    1;
  [7] 0,   705600,   705600,   201040,   24640,  1568,   56,  1;
  [8] 0, 43751232, 43751232, 12468960, 1531152, 98448, 3696, 84, 1;
		

Links

• I. M. Gessel and X. G. Viennot, Determinants, Paths, and Plane Partitions, 1989 preprint.

Crossrefs

Cf. A354042, A354045 (row sums).

Programs

Maple

T := (n, k) -> ifelse(n = 0, 1, (-1)^n*((n + 1)!/k!)*add(binomial(2*k - 2*j, k + 1)*binomial(2*n + 1, 2*j + 1)*bernoulli(2*n - 2*j) / (j - k), j = 0..(k-1)/2)): for n from 0 to 8 do seq(T(n, k), k = 0..n) od;