A229417 T(n,k) = number of n X n 0..k zero-diagonal arrays with corresponding row and column sums equal.

1, 1, 2, 1, 3, 10, 1, 4, 45, 152, 1, 5, 136, 4743, 7736, 1, 6, 325, 59008, 3801411, 1375952, 1, 7, 666, 426425, 345706336, 23938685973, 877901648, 1, 8, 1225, 2164680, 11782824375, 28256240134144, 1215663478473627, 2046320373120, 1, 9, 2080

Offset: 1

Author

R. H. Hardin, Sep 22 2013

Keywords

• nonn
• tabl

Comments

Table starts

.........1................1....................1................1............1

.........2................3....................4................5............6

........10...............45..................136..............325..........666

.......152.............4743................59008...........426425......2164680

......7736..........3801411............345706336......11782824375.213067487016

...1375952......23938685973.......28256240134144.7093199984236625

.877901648.1215663478473627.33097994593655140864

Examples

			Some solutions for n=4 k=4
..0..0..2..0....0..1..0..4....0..0..1..3....0..1..1..4....0..1..1..0
..1..0..2..1....2..0..4..0....1..0..2..3....4..0..2..3....0..0..1..2
..1..2..0..4....2..4..0..2....2..3..0..1....1..4..0..1....0..0..0..4
..0..2..3..0....1..1..4..0....1..3..3..0....1..4..3..0....2..2..2..0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..43

Crossrefs

Columns 1..3 are A007080, A229415, A229416.

Rows 3..6 are A037270(n+1), A229418, A229419, A229420.

Cf. A229870.

Formula

Empirical for row n:

n=1: a(n) = 1

n=2: a(n) = n + 1

n=3: a(n) = (1/2)*n^4 + 2*n^3 + (7/2)*n^2 + 3*n + 1

n=4: [polynomial of degree 9]

Row n is an Ehrhart polynomial of degree (n-1)^2 for the polytope of x(i,j), i,j = 1..n for j <> i, with 0 <= x(i,j) <= 1 and Sum_i x(i,j) = Sum_i x(j,i). - Robert Israel, Mar 30 2023

T(n,k) = A229870(n,k) / (k + 1)^n. - Andrew Howroyd, Mar 30 2023