A358112 - OEIS

%I #15 Nov 13 2022 14:32:57

%S 1,5,1,47,42,1,641,1659,219,1,11389,72572,28470,968,1,248749,3610485,

%T 3263402,357746,4017,1,6439075,204023334,371188155,95559940,3853617,

%U 16278,1,192621953,12989570167,43844432805,22448025251,2216662051,38270373,65399,1

%N Table read by rows. A statistic of permutations of the multiset {1,1,2,2,...,n,n}.

%C Table 1, page 12 in Maazouz and Pitman (note a typo in T(2, 0)).

%H Yassine El Maazouz and Jim Pitman, <a href="https://doi.org/10.48550/arxiv.2210.02027">The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution</a>, arXiv:2210.02027 [math.PR], Oct. 2022.

%F T(n, k) = P(n, k+1) - P(n, k), where P(n, x) = (2*n)!*Sum_{k=0..n} Sum_{j=0..n-k} binomial(n, k)*binomial(n-k, j)*(-1)^(n-k-j)*max(x - k, 0)^(2*n - j)/(2*n - j)!.

%e [n\d]    0            1           2           3           4           5     6

%e -----------------------------------------------------------------------------

%e [1]         1;

%e [2]         5,           1;

%e [3]        47,          42,           1;

%e [4]       641,        1659,         219,           1;

%e [5]     11389,       72572,       28470,         968,          1;

%e [6]    248749,     3610485,     3263402,      357746,       4017,        1;

%e [7]   6439075,   204023334,   371188155,    95559940,    3853617,    16278, 1

%e [8] 192621953, 12989570167, 43844432805, 22448025251, 2216662051, 38270373,

%e 65399, 1

%p P := (n, x) -> (2*n)!*add(add(binomial(n, k)*binomial(n-k, j)*

%p (-1)^(n-k-j)*max(x - k, 0)^(2*n - j)/(2*n - j)!, j = 0..n-k), k = 0..n):

%p Trow := n -> seq(P(n, k+1) - P(n, k), k = 0..n-1):

%p seq(print(Trow(n)), n = 1..8);

%Y Cf. A006902 (row 0), A000680 (row sums).

%K nonn,tabl

%O 1,2

%A _Peter Luschny_, Oct 30 2022