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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 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). 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.)