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A249042 Three-dimensional array of numbers N(r,p,m) read by triangular slices, each slice being read across rows: N(r,p,m) is the number of "r-panes in a (p,m) structure".

%I A249042 #27 Nov 01 2018 13:38:22
%S A249042 1,1,1,2,1,3,4,1,6,6,1,6,7,7,24,18,1,14,36,24,1,10,11,25,70,46,15,100,
%T A249042 180,96,1,30,150,240,120,1,15,16,65,165,101,90,455,690,326,31,360,
%U A249042 1170,1440,600,1,62,540,1560,1800,720
%N A249042 Three-dimensional array of numbers N(r,p,m) read by triangular slices, each slice being read across rows: N(r,p,m) is the number of "r-panes in a (p,m) structure".
%C A249042 Three-dimensional arrays don't really work in the OEIS, but this one seems like it should be included. See Good-Tideman for precise definition.
%H A249042 I. J. Good, T. N. Tideman, <a href="http://dx.doi.org/10.1016/0097-3165(77)90077-2">Stirling numbers and a geometric structure from voting theory</a>, Journal of Combinatorial Theory, Series A Volume 23, Issue 1, July 1977, Pages 34-45.
%H A249042 Warren D. Smith, <a href="http://rangevoting.org/WilsonOrder.html">D-dimensional orderings and Stirling numbers</a>, October 2014.
%F A249042 There is a formula involving Stirling numbers.
%e A249042 The initial triangular slices are:
%e A249042 1
%e A249042 -
%e A249042 1
%e A249042 1 2
%e A249042 ---
%e A249042 1
%e A249042 3 4
%e A249042 1 6 6
%e A249042 -----
%e A249042 1
%e A249042 6 7
%e A249042 7 24 18
%e A249042 1 14 36 24
%e A249042 ----------
%e A249042 1
%e A249042 10 11
%e A249042 25 70  46
%e A249042 15 100 180 96
%e A249042 1  30  150 240 120
%e A249042 ----------------
%e A249042 1
%e A249042 15 16
%e A249042 65 165 101
%e A249042 90 455 690  326
%e A249042 31 360 1170 1440 600
%e A249042 1  62  540  1560 1800 720
%t A249042 S1[m_, n_] := Abs[StirlingS1[m, m - n]];
%t A249042 S2[m_, n_] := StirlingS2[m, m - n];
%t A249042 Nr[r_, p_, m_] := S2[m, p - r] Sum[S1[m - p + r, nu], {nu, 0, r}];
%t A249042 Table[Nr[r, p, m], {m, 1, 6}, {p, 0, m - 1}, {r, 0, p}] // Flatten (* _Jean-François Alcover_, Nov 01 2018 *)
%Y A249042 The sequence of left edges of the triangles is A008278; the bases of the triangles give A019538; the hypotenuses give A181854.
%K A249042 nonn,tabf,more
%O A249042 1,4
%A A249042 _N. J. A. Sloane_, Oct 29 2014
%E A249042 More terms from _Michel Marcus_, Aug 28 2015