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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.

%I A354043 #14 Dec 26 2024 23:03:13
%S A354043 1,0,1,0,1,1,0,4,4,1,0,36,36,10,1,0,600,600,170,20,1,0,16584,16584,
%T A354043 4720,574,35,1,0,705600,705600,201040,24640,1568,56,1,0,43751232,
%U A354043 43751232,12468960,1531152,98448,3696,84,1,0,3790108800,3790108800,1080240480,132713280,8554896,325152,7812,120,1
%N A354043 Table read by rows: T(n, k) = (-1)^(n-k)*F(n, k)/k!, where F are the Faulhaber numbers A354042.
%C A354043 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:
%C A354043 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.
%C A354043 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.
%H A354043 I. M. Gessel and X. G. Viennot, <a href="https://people.brandeis.edu/~gessel/homepage/papers/pp.pdf">Determinants, Paths, and Plane Partitions</a>, 1989 preprint.
%e A354043 Table starts:
%e A354043   [0] 1;
%e A354043   [1] 0,        1;
%e A354043   [2] 0,        1,        1;
%e A354043   [3] 0,        4,        4,        1;
%e A354043   [4] 0,       36,       36,       10,       1;
%e A354043   [5] 0,      600,      600,      170,      20,     1;
%e A354043   [6] 0,    16584,    16584,     4720,     574,    35,    1;
%e A354043   [7] 0,   705600,   705600,   201040,   24640,  1568,   56,  1;
%e A354043   [8] 0, 43751232, 43751232, 12468960, 1531152, 98448, 3696, 84, 1;
%p A354043 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;
%Y A354043 Cf. A354042, A354045 (row sums).
%K A354043 nonn,tabl
%O A354043 0,8
%A A354043 _Peter Luschny_, May 17 2022