A092477 Triangle read by rows: T(n,k) = (2^k - 1)^n, 1<=k<=n.

1, 1, 9, 1, 27, 343, 1, 81, 2401, 50625, 1, 243, 16807, 759375, 28629151, 1, 729, 117649, 11390625, 887503681, 62523502209, 1, 2187, 823543, 170859375, 27512614111, 3938980639167, 532875860165503, 1, 6561, 5764801, 2562890625, 852891037441, 248155780267521, 67675234241018881, 17878103347812890625

Offset: 1

Author

Reinhard Zumkeller, Mar 26 2004

Keywords

• nonn
• tabl

Comments

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

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]

From Manfred Boergens, Jun 23 2024: (Start)

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.

For nonempty A_j see A218695.

For disjoint A_j see A089072.

For nonempty and disjoint A_j see A019538.

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

Examples

			Triangle begins
 1
 1,9;
 1,27,343;
 1,81,2401,50625;
 1,243,16807,759375, 28629151 [_Geoffrey Critzer_, Dec 03 2009]
		

Crossrefs

Cf. A019538, A089072, A218695, A329943.

Programs

Maple

A092477 := proc(n,k)
    (2^k-1)^n ;
end proc:
seq(seq( A092477(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Nov 18 2023

Mathematica

Table[Table[(2^k - 1)^n, {k, 1, n}], {n, 1, 6}] // Grid (* Geoffrey Critzer, Dec 03 2009 *)

Extensions

More terms from Michel Marcus, Jun 23 2024