cp's OEIS Frontend

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

%I A145836 #19 Aug 10 2018 02:34:32
%S A145836 0,0,0,0,0,0,0,0,0,0,0,0,10080,0,0,0,15120,544320,544320,15120,0,0,
%T A145836 40320,1958040,6108480,1958040,40320,0,0,24192,1796760,12267360,
%U A145836 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 A145836 Coefficients of a symmetric matrix representation of the 9th falling factorial power, read by antidiagonals.
%C A145836 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 A145836 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 A145836 Full array of coefficients:
%e A145836 [0,     0,       0,        0,        0,       0,      0,       0,    1],
%e A145836 [0,     0,       0,        0,    15120,   40320,   24192,   4608,  255],
%e A145836 [0,     0,   10080,   544320,  1958040, 1796760,  588168,  74124, 3025],
%e A145836 [0,     0,  544320,  6108480, 12267360, 7988904, 2066232, 218484, 7770],
%e A145836 [0, 15120, 1958040, 12267360, 18329850, 9874746, 2229402, 212436, 6951],
%e A145836 [0, 40320, 1796760,  7988904,  9874746, 4690350,  965790,  85680, 2646],
%e A145836 [0, 24192,  588168,  2066232,  2229402,  965790,  185766,  15624,  462],
%e A145836 [0,  4608,   74124,   218484,   212436,   85680,   15624,   1260,   36],
%e A145836 [1,   255,    3025,     7770,     6951,    2646,     462,     36,    1]
%t A145836 rows = 9;
%t A145836 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 A145836 c[rows, _, _] = Nothing;
%t A145836 Table[Table[c[rows, l-m+1, m], {m, 1, l}], {l, 1, 2rows-1}] // Flatten (* _Jean-François Alcover_, Aug 10 2018 *)
%Y A145836 Cf. A008277, A068424.
%K A145836 fini,full,nonn
%O A145836 0,13
%A A145836 _Jonathan Vos Post_, Oct 21 2008
%E A145836 Corrected by _Michel Marcus_, Dec 15 2014