A145836 Coefficients of a symmetric matrix representation of the 9th falling factorial power, read by antidiagonals.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10080, 0, 0, 0, 15120, 544320, 544320, 15120, 0, 0, 40320, 1958040, 6108480, 1958040, 40320, 0, 0, 24192, 1796760, 12267360, 12267360, 1796760, 24192, 0, 1, 4608, 588168, 7988904, 18329850, 7988904, 588168, 4608, 1, 255, 74124, 2066232, 9874746, 9874746, 2066232, 74124, 255, 3025, 218484, 2229402, 4690350, 2229402, 218484, 3025, 7770, 212436, 965790, 965790, 212436, 7770, 6951, 85680, 185766, 85680, 6951, 2646, 15624, 15624, 2646, 462, 1260, 462, 36, 36, 1

Offset: 0

Author

Jonathan Vos Post, Oct 21 2008

Keywords

• fini
• full
• nonn

Comments

Osgood and Wu abstract: We investigate the coefficients generated by expressing the falling factorial (xy)_k as a linear combination of falling factorial products (x)_l (y)_m for l,m = 1,...,k. Algebraic and combinatoric properties of these coefficients are discussed, including recurrence relations, closed-form formulas, relations with Stirling numbers and a combinatorial characterization in terms of conjoint ranking tables.

Examples

			Full array of coefficients:
[0,     0,       0,        0,        0,       0,      0,       0,    1],
[0,     0,       0,        0,    15120,   40320,   24192,   4608,  255],
[0,     0,   10080,   544320,  1958040, 1796760,  588168,  74124, 3025],
[0,     0,  544320,  6108480, 12267360, 7988904, 2066232, 218484, 7770],
[0, 15120, 1958040, 12267360, 18329850, 9874746, 2229402, 212436, 6951],
[0, 40320, 1796760,  7988904,  9874746, 4690350,  965790,  85680, 2646],
[0, 24192,  588168,  2066232,  2229402,  965790,  185766,  15624,  462],
[0,  4608,   74124,   218484,   212436,   85680,   15624,   1260,   36],
[1,   255,    3025,     7770,     6951,    2646,     462,     36,    1]
		

Links

• Brad Osgood, William Wu, Falling Factorials, Generating Functions and Conjoint Ranking Tables, arXiv:0810.3327 [math.CO], 2008.

Crossrefs

Cf. A008277, A068424.

Programs

Mathematica

rows = 9;
c[k_, l_ /; l <= rows, m_ /; m <= rows] := Sum[(-1)^(k-p) Abs[StirlingS1[k, p]] StirlingS2[p, l] StirlingS2[p, m], {p, 1, k}];
c[rows, , ] = Nothing;
Table[Table[c[rows, l-m+1, m], {m, 1, l}], {l, 1, 2rows-1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

Extensions

Corrected by Michel Marcus, Dec 15 2014