A051129 Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).

1, 1, 2, 1, 4, 3, 1, 8, 9, 4, 1, 16, 27, 16, 5, 1, 32, 81, 64, 25, 6, 1, 64, 243, 256, 125, 36, 7, 1, 128, 729, 1024, 625, 216, 49, 8, 1, 256, 2187, 4096, 3125, 1296, 343, 64, 9, 1, 512, 6561, 16384, 15625, 7776, 2401, 512, 81, 10, 1, 1024, 19683, 65536, 78125, 46656, 16807, 4096, 729, 100, 11

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy
• nice

Comments

(n-th term) = (n-th term of A002260)^(n-th term of A004736). Both A002260 and A004736 are related to A002024. - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

Examples

			  1   2       3       4       5       6       7
  1   4       9      16      25      36      49
  1   8      27      64     125     216     343
  1  16      81     256     625    1296    2401
  1  32     243    1024    3125    7776   16807
  1  64     729    4096   15625   46656  117649
  1 128    2187   16384   78125  279936  823543
		

Links

• T. D. Noe, Rows n = 1..50 of triangle, flattened

Crossrefs

Cf. A051128 (transposed), A003992 (transposed), A004248.

Cf. A002024, A002260, A004736.

Cf. A002260, A003101 (antidiagonal sums), A000169 (central terms), A003320 (row maxima), A247358 (sorted rows).

Programs

Haskell

a051129 n k = k ^ (n - k)
a051129_row n = a051129_tabl !! (n-1)
a051129_tabl = zipWith (zipWith (^)) a002260_tabl $ map reverse a002260_tabl
-- Reinhard Zumkeller, Sep 14 2014

Maple

T:= (n, k)-> k^n:
seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # Alois P. Heinz, Apr 18 2020

Mathematica

Table[ k^(n-k+1), {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 30 2012 *)

PARI

b(n) = floor(1/2 + sqrt(2 * n));
vector(100, n, (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1)) \\ Altug Alkan, Dec 09 2015

Formula

a(n) = (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1), where b(n) = [ 1/2 + sqrt(2 * n) ]. (b(n) is the n-th term of A002024.) - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

Extensions

More terms from James Sellers, Dec 11 1999