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

0,8

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.

Links

Table of n, a(n) for n=0..54.

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

Example

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;

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;

Crossrefs

Cf. A354042, A354045 (row sums).

Sequence in context: A164612 A309748 A180401 * A057270 A057278 A010303

Adjacent sequences: A354040 A354041 A354042 * A354044 A354045 A354046

Keyword

nonn,tabl

Author

Peter Luschny, May 17 2022

Status

approved