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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, 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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Aug 10 2018 *)

CROSSREFS

Cf. A008277, A068424.

Sequence in context: A252200 A234773 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

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Last modified November 17 07:44 EST 2018. Contains 317275 sequences. (Running on oeis4.)