A245789 Rectangular array A read by upward antidiagonals: A(k,n) = (2^k-1)^n, n,k >= 1.

1, 1, 3, 1, 9, 7, 1, 27, 49, 15, 1, 81, 343, 225, 31, 1, 243, 2401, 3375, 961, 63, 1, 729, 16807, 50625, 29791, 3969, 127, 1, 2187, 117649, 759375, 923521, 250047, 16129, 255, 1, 6561, 823543, 11390625, 28629151, 15752961, 2048383, 65025, 511

Offset: 1

Author

L. Edson Jeffery, Aug 22 2014

Keywords

• nonn
• tabl

Comments

A(k,n) is the number of sequences (X_1, X_2, ..., X_k) of subsets of the set {1, 2, ..., n} such that intersect_{j=1..k} X_j = null.

Examples

			Array A begins:
1      3         7           15              31                 63
1      9        49          225             961               3969
1     27       343         3375           29791             250047
1     81      2401        50625          923521           15752961
1    243     16807       759375        28629151          992436543
1    729    117649     11390625       887503681        62523502209
1   2187    823543    170859375     27512614111      3938980639167
1   6561   5764801   2562890625    852891037441    248155780267521
1  19683  40353607  38443359375  26439622160671  15633814156853823
		

References

• Richard P. Stanley, Enumerative Combinatorics, Cambridge University Press, Vol. 1, Second edition, 2012, p. 14 (Example 1.1.16).

Links

• L. Edson Jeffery, Table of n, a(n) for n = 1..45

Crossrefs

Cf. A000225, A060867, A128831, etc. (rows 1-3).

Cf. A000012, A000244, A000420, etc. (columns 1-3).

Cf. A055601 (main diagonal).

Programs

Mathematica

(* Array *)
a[k_, n_] := (2^k - 1)^n; Grid[Table[a[k, n], {n, 12}, {k, 12}]]
(* Array antidiagonals flattened *)
Flatten[Table[(2^k - 1)^(n - k + 1), {n, 12}, {k, n}]]