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A051129 Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1). 18

%I #39 Oct 09 2023 01:44:30

%S 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 7,1,128,729,1024,625,216,49,8,1,256,2187,4096,3125,1296,343,64,9,1,

%U 512,6561,16384,15625,7776,2401,512,81,10,1,1024,19683,65536,78125,46656,16807,4096,729,100,11

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

%C (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 T. D. Noe, <a href="/A051129/b051129.txt">Rows n = 1..50 of triangle, flattened</a>

%F 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 1 2 3 4 5 6 7

%e 1 4 9 16 25 36 49

%e 1 8 27 64 125 216 343

%e 1 16 81 256 625 1296 2401

%e 1 32 243 1024 3125 7776 16807

%e 1 64 729 4096 15625 46656 117649

%e 1 128 2187 16384 78125 279936 823543

%p T:= (n, k)-> k^n:

%p seq(seq(T(1+d-k, k), k=1..d), d=1..11); # _Alois P. Heinz_, Apr 18 2020

%t Table[ k^(n-k+1), {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 30 2012 *)

%o (Haskell)

%o a051129 n k = k ^ (n - k)

%o a051129_row n = a051129_tabl !! (n-1)

%o a051129_tabl = zipWith (zipWith (^)) a002260_tabl $ map reverse a002260_tabl

%o -- _Reinhard Zumkeller_, Sep 14 2014

%o (PARI) b(n) = floor(1/2 + sqrt(2 * n));

%o vector(100, n, (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1)) \\ _Altug Alkan_, Dec 09 2015

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

%Y Cf. A002024, A002260, A004736.

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

%K nonn,tabl,easy,nice

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Dec 11 1999

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Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)