A358112 Table read by rows. A statistic of permutations of the multiset {1,1,2,2,...,n,n}.

1, 5, 1, 47, 42, 1, 641, 1659, 219, 1, 11389, 72572, 28470, 968, 1, 248749, 3610485, 3263402, 357746, 4017, 1, 6439075, 204023334, 371188155, 95559940, 3853617, 16278, 1, 192621953, 12989570167, 43844432805, 22448025251, 2216662051, 38270373, 65399, 1

Offset: 1

Author

Peter Luschny, Oct 30 2022

Keywords

• nonn
• tabl

Comments

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

Examples

			[n\d]    0            1           2           3           4           5     6
-----------------------------------------------------------------------------
[1]         1;
[2]         5,           1;
[3]        47,          42,           1;
[4]       641,        1659,         219,           1;
[5]     11389,       72572,       28470,         968,          1;
[6]    248749,     3610485,     3263402,      357746,       4017,        1;
[7]   6439075,   204023334,   371188155,    95559940,    3853617,    16278, 1
[8] 192621953, 12989570167, 43844432805, 22448025251, 2216662051, 38270373,
65399, 1
		

Links

• Yassine El Maazouz and Jim Pitman, The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution, arXiv:2210.02027 [math.PR], Oct. 2022.

Crossrefs

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

Programs

Maple

P := (n, x) -> (2*n)!*add(add(binomial(n, k)*binomial(n-k, j)*
(-1)^(n-k-j)*max(x - k, 0)^(2*n - j)/(2*n - j)!, j = 0..n-k), k = 0..n):
Trow := n -> seq(P(n, k+1) - P(n, k), k = 0..n-1):
seq(print(Trow(n)), n = 1..8);

Formula

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)!.