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%I #34 Apr 27 2024 18:01:02
%S 1,0,1,0,1,1,0,1,2,1,0,1,4,3,1,0,1,8,9,4,1,0,1,16,27,16,5,1,0,1,32,81,
%T 64,25,6,1,0,1,64,243,256,125,36,7,1,0,1,128,729,1024,625,216,49,8,1,
%U 0,1,256,2187,4096,3125,1296,343,64,9,1,0,1,512,6561,16384,15625,7776,2401,512,81,10,1
%N Array read by ascending antidiagonals: A(n, k) = k^n.
%C This array transforms into A371761 using the Akiyama-Tanigawa algorithm for powers applied to the rows. - _Peter Luschny_, Apr 16 2024
%C This array transforms into A344499 using the Akiyama-Tanigawa algorithm for powers applied to the columns. - _Peter Luschny_, Apr 27 2024
%H T. D. Noe, <a href="/A004248/b004248.txt">Rows n=0..50 of triangle, flattened</a>
%H Michelle Rudolph-Lilith, <a href="http://arxiv.org/abs/1508.07894">On the Product Representation of Number Sequences, with Application to the Fibonacci Family</a>, arXiv preprint arXiv:1508.07894 [math.NT], 2015.
%F Table of x^y, where (x,y) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...
%F As a number triangle, columns have g.f. x^k/(1 - kx). - _Paul Barry_, Mar 28 2005
%F From _Paul Barry_, Jul 13 2005: (Start)
%F T(n, k) = if(k <= n, k^(n - k), 0).
%F T(n, k) = Sum_{j=0..floor((n-k)/2)} (-1)^j*C(n-k, j)*C(n-k-j, n-k)*k^(n-k-2j).
%F (End)
%e Seen as an array that is read by ascending antidiagonals:
%e [0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e [1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
%e [2] 0, 1, 4, 9, 16, 25, 36, 49, 64, ...
%e [3] 0, 1, 8, 27, 64, 125, 216, 343, 512, ...
%e [4] 0, 1, 16, 81, 256, 625, 1296, 2401, 4096, ...
%e [5] 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, ...
%e [6] 0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, ...
%e [7] 0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, ...
%t T[x_, y_] := If[y == 0, 1, (x - y)^y];
%t Table[T[x, y], {x, 0, 11}, {y, x, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 15 2017 *)
%o (PARI)
%o T(x, y) = x^y \\ _Charles R Greathouse IV_, Feb 07 2017
%o (SageMath)
%o def Arow(n, len): return [k**n for k in range(len)]
%o for n in range(8): print([n], Arow(n, 9)) # _Peter Luschny_, Apr 16 2024
%Y For other versions see A051129 and A009998.
%Y Row sums are A026898, diagonal sums are A104872. [_Paul Barry_, Mar 28 2005]
%Y Cf. A003992, A048723, A371761, A344499.
%K tabl,nonn,easy,nice,changed
%O 0,9
%A _David W. Wilson_
%E New name by _Peter Luschny_, Apr 16 2024.
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