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A354043 Table read by rows: T(n, k) = (-1)^(n-k)*F(n, k)/k!, where F are the Faulhaber numbers A354042. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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} ยทยทยท 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

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Last modified August 15 00:08 EDT 2022. Contains 356122 sequences. (Running on oeis4.)