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".

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, 180, 96, 1, 30, 150, 240, 120, 1, 15, 16, 65, 165, 101, 90, 455, 690, 326, 31, 360, 1170, 1440, 600, 1, 62, 540, 1560, 1800, 720

Offset: 1

Author

N. J. A. Sloane, Oct 29 2014

Keywords

• nonn
• tabf
• more

Comments

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.

Examples

			The initial triangular slices are:
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
15 100 180 96
1  30  150 240 120
----------------
1
15 16
65 165 101
90 455 690  326
31 360 1170 1440 600
1  62  540  1560 1800 720
		

Links

• I. J. Good, T. N. Tideman, Stirling numbers and a geometric structure from voting theory, Journal of Combinatorial Theory, Series A Volume 23, Issue 1, July 1977, Pages 34-45.
• Warren D. Smith, D-dimensional orderings and Stirling numbers, October 2014.

Crossrefs

The sequence of left edges of the triangles is A008278; the bases of the triangles give A019538; the hypotenuses give A181854.

Programs

Mathematica

S1[m_, n_] := Abs[StirlingS1[m, m - n]];
S2[m_, n_] := StirlingS2[m, m - n];
Nr[r_, p_, m_] := S2[m, p - r] Sum[S1[m - p + r, nu], {nu, 0, r}];
Table[Nr[r, p, m], {m, 1, 6}, {p, 0, m - 1}, {r, 0, p}] // Flatten (* Jean-François Alcover, Nov 01 2018 *)

Formula

There is a formula involving Stirling numbers.

Extensions

More terms from Michel Marcus, Aug 28 2015