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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). 0
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 (list; graph; refs; listen; history; text; internal format)
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 A349342

Adjacent sequences:  A335153 A335154 A335155 * A335157 A335158 A335159

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Jun 06 2020

STATUS

approved

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Last modified December 1 20:25 EST 2021. Contains 349435 sequences. (Running on oeis4.)