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

0,13

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.

Links

Table of n, a(n) for n=0..80.

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

Example

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]

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 *)

Crossrefs

Cf. A008277, A068424.

Sequence in context: A234773 A348698 A203865 * A190293 A179729 A217867

Adjacent sequences: A145833 A145834 A145835 * A145837 A145838 A145839

Keyword

fini,full,nonn

Author

Jonathan Vos Post, Oct 21 2008

Extensions

Corrected by Michel Marcus, Dec 15 2014

Status

approved