A335156 Three-dimensional array A(m;r,s) = Sum_{i=0..r} Sum_{j=0..s} (-1)^(i+j+r+s)*binomial(r,i)*binomial(s,j)*binomial(i*j,m) displayed as a series of square slices read across rows (see Comments for details).

1, 0, 1, 1, 2, 0, 0, 1, 0, 4, 6, 1, 6, 6, 0, 0, 0, 1, 0, 1, 12, 14, 0, 12, 45, 36, 1, 14, 36, 24, 0, 0, 0, 0, 1, 0, 0, 6, 32, 30, 0, 6, 90, 228, 150, 0, 32, 228, 432, 240, 1, 30, 150, 240, 120, 0, 0, 0, 0, 0, 1, 0, 0, 1, 24, 80, 62, 0, 1, 78, 522, 975, 540, 0, 24, 522, 2248, 3300, 1560, 0, 80, 975, 3300, 4200, 1800, 1, 62, 540, 1560, 1800, 720

Offset

1,5

Comments

This 3-D array is represented as a triangle in which row m shows the square matrix A(m;r,s) (1 <= r <= m, 1 <= s <= m) read across rows.

The array is symmetrical in the sense that f(m;r,s) = f(m;s,r).

Links

Table of n, a(n) for n=1..91.

John Riordan and Paul R. Stein, Arrangements on chessboards, Journal of Combinatorial Theory, Series A 12.1 (1972): 72-80. See A(m;r,s) in Section 2, Eq. (9).

Index entries for sequences related to 3D arrays of numbers

Example

The square slices for m = 1,2,3,4,5,6,7 are:
m=1:
[[1]],
m=2:
[[0, 1],
[1, 2]],
m=3:
[[0, 0, 1],
[0, 4, 6],
[1, 6, 6]],
m=4:
[[0, 0, 0, 1],
[0, 1, 12, 14],
[0, 12, 45, 36],
[1, 14, 36, 24]],
m=5:
[[0, 0, 0, 0, 1],
[0, 0, 6, 32, 30],
[0, 6, 90, 228, 150],
[0, 32, 228, 432, 240],
[1, 30, 150, 240, 120]],
m=6:
[[0, 0, 0, 0, 0, 1],
[0, 0, 1, 24, 80, 62],
[0, 1, 78, 522, 975, 540],
[0, 24, 522, 2248, 3300, 1560],
[0, 80, 975, 3300, 4200, 1800],
[1, 62, 540, 1560, 1800, 720]],
m=7:
[[0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 8, 80, 192, 126],
[0, 0, 36, 624, 2595, 3798, 1806],
[0, 8, 624, 5776, 17300, 20520, 8400],
[0, 80, 2595, 17300, 43000, 45000, 16800],
[0, 192, 3798, 20520, 45000, 43200, 15120],
[1, 126, 1806, 8400, 16800, 15120, 5040]],

Maple

f:=(m, r, s) -> add( add( (-1)^(i+j+r+s)*binomial(r, i)*binomial(s, j)*binomial(i*j, m), j=0..s), i=0..r);
sq:=m -> [seq([seq(f(m, r, s), r=1..m)], s=1..m)];
for m from 1 to 8 do lprint(sq(m)); od:

Crossrefs

Sequence in context: A347239 A347097 A339274 * A158785 A346243 A349916

Adjacent sequences: A335153 A335154 A335155 * A335157 A335158 A335159

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jun 06 2020

Status

approved