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

%I #30 Nov 30 2014 15:13:49

%S 1,1,3,1,9,7,1,27,49,15,1,81,343,225,31,1,243,2401,3375,961,63,1,729,

%T 16807,50625,29791,3969,127,1,2187,117649,759375,923521,250047,16129,

%U 255,1,6561,823543,11390625,28629151,15752961,2048383,65025,511

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

%C 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.

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

%H L. Edson Jeffery, <a href="/A245789/b245789.txt">Table of n, a(n) for n = 1..45</a>

%e Array A begins:

%e 1 3 7 15 31 63

%e 1 9 49 225 961 3969

%e 1 27 343 3375 29791 250047

%e 1 81 2401 50625 923521 15752961

%e 1 243 16807 759375 28629151 992436543

%e 1 729 117649 11390625 887503681 62523502209

%e 1 2187 823543 170859375 27512614111 3938980639167

%e 1 6561 5764801 2562890625 852891037441 248155780267521

%e 1 19683 40353607 38443359375 26439622160671 15633814156853823

%t (* Array *)

%t a[k_, n_] := (2^k - 1)^n; Grid[Table[a[k, n], {n, 12}, {k, 12}]]

%t (* Array antidiagonals flattened *)

%t Flatten[Table[(2^k - 1)^(n - k + 1), {n, 12}, {k, n}]]

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

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

%Y Cf. A055601 (main diagonal).

%K nonn,tabl

%O 1,3

%A _L. Edson Jeffery_, Aug 22 2014

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Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)