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Triangle read by rows: T(n,k) = (2^k - 1)^n, 1<=k<=n.
10

%I #14 Jun 23 2024 11:51:51

%S 1,1,9,1,27,343,1,81,2401,50625,1,243,16807,759375,28629151,1,729,

%T 117649,11390625,887503681,62523502209,1,2187,823543,170859375,

%U 27512614111,3938980639167,532875860165503,1,6561,5764801,2562890625,852891037441,248155780267521,67675234241018881,17878103347812890625

%N Triangle read by rows: T(n,k) = (2^k - 1)^n, 1<=k<=n.

%C T(n,1)=1; T(n,2)=A000244(n); T(n,n-1)=A086206(n); T(n,n)=A055601(n).

%C T(n,k) is the number of n X k binary matrices with no 0 rows. The triangular array becomes a rectangular array by lifting the restriction on k. [From _Geoffrey Critzer_, Dec 03 2009]

%C From _Manfred Boergens_, Jun 23 2024: (Start)

%C T(n,k) is the number of coverings of [n] by tuples (A_1,...,A_k) in P([n])^k, with P(.) denoting the power set.

%C For nonempty A_j see A218695.

%C For disjoint A_j see A089072.

%C For nonempty and disjoint A_j see A019538.

%C Lifting the restriction on k and swapping n,k gives A329943. (End)

%e Triangle begins

%e 1

%e 1,9;

%e 1,27,343;

%e 1,81,2401,50625;

%e 1,243,16807,759375, 28629151 [_Geoffrey Critzer_, Dec 03 2009]

%p A092477 := proc(n,k)

%p (2^k-1)^n ;

%p end proc:

%p seq(seq( A092477(n,k),k=1..n),n=1..12) ; # _R. J. Mathar_, Nov 18 2023

%t Table[Table[(2^k - 1)^n, {k, 1, n}], {n, 1, 6}] // Grid (* _Geoffrey Critzer_, Dec 03 2009 *)

%Y Cf. A019538, A089072, A218695, A329943.

%K nonn,tabl

%O 1,3

%A _Reinhard Zumkeller_, Mar 26 2004

%E More terms from _Michel Marcus_, Jun 23 2024