login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

%I

%S 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,

%T 0,0,0,0,1,0,0,6,32,30,0,6,90,228,150,0,32,228,432,240,1,30,150,240,

%U 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

%N 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).

%C 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.

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

%H John Riordan and Paul R. Stein, <a href="https://doi.org/10.1016/0097-3165(72)90084-2">Arrangements on chessboards</a>, Journal of Combinatorial Theory, Series A 12.1 (1972): 72-80. See A(m;r,s) in Section 2, Eq. (9).

%H <a href="/index/3#3DArrays">Index entries for sequences related to 3D arrays of numbers</a>

%e The square slices for m = 1,2,3,4,5,6,7 are:

%e m=1:

%e [[1]],

%e m=2:

%e [[0, 1],

%e [1, 2]],

%e m=3:

%e [[0, 0, 1],

%e [0, 4, 6],

%e [1, 6, 6]],

%e m=4:

%e [[0, 0, 0, 1],

%e [0, 1, 12, 14],

%e [0, 12, 45, 36],

%e [1, 14, 36, 24]],

%e m=5:

%e [[0, 0, 0, 0, 1],

%e [0, 0, 6, 32, 30],

%e [0, 6, 90, 228, 150],

%e [0, 32, 228, 432, 240],

%e [1, 30, 150, 240, 120]],

%e m=6:

%e [[0, 0, 0, 0, 0, 1],

%e [0, 0, 1, 24, 80, 62],

%e [0, 1, 78, 522, 975, 540],

%e [0, 24, 522, 2248, 3300, 1560],

%e [0, 80, 975, 3300, 4200, 1800],

%e [1, 62, 540, 1560, 1800, 720]],

%e m=7:

%e [[0, 0, 0, 0, 0, 0, 1],

%e [0, 0, 0, 8, 80, 192, 126],

%e [0, 0, 36, 624, 2595, 3798, 1806],

%e [0, 8, 624, 5776, 17300, 20520, 8400],

%e [0, 80, 2595, 17300, 43000, 45000, 16800],

%e [0, 192, 3798, 20520, 45000, 43200, 15120],

%e [1, 126, 1806, 8400, 16800, 15120, 5040]],

%p 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);

%p sq:=m -> [seq([seq(f(m,r,s),r=1..m)],s=1..m)];

%p for m from 1 to 8 do lprint(sq(m)); od:

%K nonn,tabf

%O 1,5

%A _N. J. A. Sloane_, Jun 06 2020

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 28 04:25 EST 2022. Contains 350654 sequences. (Running on oeis4.)