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

%I A051129 #40 Jul 02 2025 16:01:58
%S A051129 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,
%T A051129 7,1,128,729,1024,625,216,49,8,1,256,2187,4096,3125,1296,343,64,9,1,
%U A051129 512,6561,16384,15625,7776,2401,512,81,10,1,1024,19683,65536,78125,46656,16807,4096,729,100,11
%N A051129 Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).
%C A051129 (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
%H A051129 T. D. Noe, <a href="/A051129/b051129.txt">Rows n = 1..50 of triangle, flattened</a>
%F A051129 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
%e A051129   1   2       3       4       5       6       7
%e A051129   1   4       9      16      25      36      49
%e A051129   1   8      27      64     125     216     343
%e A051129   1  16      81     256     625    1296    2401
%e A051129   1  32     243    1024    3125    7776   16807
%e A051129   1  64     729    4096   15625   46656  117649
%e A051129   1 128    2187   16384   78125  279936  823543
%p A051129 T:= (n, k)-> k^n:
%p A051129 seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # _Alois P. Heinz_, Apr 18 2020
%t A051129 Table[ k^(n-k+1), {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 30 2012 *)
%o A051129 (Haskell)
%o A051129 a051129 n k = k ^ (n - k)
%o A051129 a051129_row n = a051129_tabl !! (n-1)
%o A051129 a051129_tabl = zipWith (zipWith (^)) a002260_tabl $ map reverse a002260_tabl
%o A051129 -- _Reinhard Zumkeller_, Sep 14 2014
%o A051129 (PARI) b(n) = floor(1/2 + sqrt(2 * n));
%o A051129 vector(100, n, (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1)) \\ _Altug Alkan_, Dec 09 2015
%Y A051129 Cf. A051128 (transposed), A003992 (transposed), A004248.
%Y A051129 Cf. A002024, A002260, A004736.
%Y A051129 Cf. A002260, A003101 (antidiagonal sums), A000169 (central terms), A003320 (row maxima), A247358 (sorted rows).
%K A051129 nonn,tabl,easy,nice
%O A051129 1,3
%A A051129 _N. J. A. Sloane_
%E A051129 More terms from _James Sellers_, Dec 11 1999