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

1,3

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

Links

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

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

Example

  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

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 *)

Prog

(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
(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

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).

Sequence in context: A247023 A356250 A348702 * A319075 A264871 A067410

Adjacent sequences: A051126 A051127 A051128 * A051130 A051131 A051132

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Dec 11 1999

Status

approved