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%I #19 Aug 10 2018 02:34:32
%S 0,0,0,0,0,0,0,0,0,0,0,0,10080,0,0,0,15120,544320,544320,15120,0,0,
%T 40320,1958040,6108480,1958040,40320,0,0,24192,1796760,12267360,
%U 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
%N Coefficients of a symmetric matrix representation of the 9th falling factorial power, read by antidiagonals.
%C 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.
%H Brad Osgood, William Wu, <a href="http://arxiv.org/abs/0810.3327">Falling Factorials, Generating Functions and Conjoint Ranking Tables</a>, arXiv:0810.3327 [math.CO], 2008.
%e Full array of coefficients:
%e [0, 0, 0, 0, 0, 0, 0, 0, 1],
%e [0, 0, 0, 0, 15120, 40320, 24192, 4608, 255],
%e [0, 0, 10080, 544320, 1958040, 1796760, 588168, 74124, 3025],
%e [0, 0, 544320, 6108480, 12267360, 7988904, 2066232, 218484, 7770],
%e [0, 15120, 1958040, 12267360, 18329850, 9874746, 2229402, 212436, 6951],
%e [0, 40320, 1796760, 7988904, 9874746, 4690350, 965790, 85680, 2646],
%e [0, 24192, 588168, 2066232, 2229402, 965790, 185766, 15624, 462],
%e [0, 4608, 74124, 218484, 212436, 85680, 15624, 1260, 36],
%e [1, 255, 3025, 7770, 6951, 2646, 462, 36, 1]
%t rows = 9;
%t 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}];
%t c[rows, _, _] = Nothing;
%t Table[Table[c[rows, l-m+1, m], {m, 1, l}], {l, 1, 2rows-1}] // Flatten (* _Jean-François Alcover_, Aug 10 2018 *)
%Y Cf. A008277, A068424.
%K fini,full,nonn
%O 0,13
%A _Jonathan Vos Post_, Oct 21 2008
%E Corrected by _Michel Marcus_, Dec 15 2014
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