OFFSET
0,9
COMMENTS
This array transforms into A371761 using the Akiyama-Tanigawa algorithm for powers applied to the rows. - Peter Luschny, Apr 16 2024
This array transforms into A344499 using the Akiyama-Tanigawa algorithm for powers applied to the columns. - Peter Luschny, Apr 27 2024
LINKS
T. D. Noe, Rows n=0..50 of triangle, flattened
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.
FORMULA
Table of x^y, where (x,y) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...
As a number triangle, columns have g.f. x^k/(1 - kx). - Paul Barry, Mar 28 2005
From Paul Barry, Jul 13 2005: (Start)
T(n, k) = if(k <= n, k^(n - k), 0).
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).
(End)
EXAMPLE
Seen as an array that is read by ascending antidiagonals:
[0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
[1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
[2] 0, 1, 4, 9, 16, 25, 36, 49, 64, ...
[3] 0, 1, 8, 27, 64, 125, 216, 343, 512, ...
[4] 0, 1, 16, 81, 256, 625, 1296, 2401, 4096, ...
[5] 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, ...
[6] 0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, ...
[7] 0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, ...
MATHEMATICA
T[x_, y_] := If[y == 0, 1, (x - y)^y];
Table[T[x, y], {x, 0, 11}, {y, x, 0, -1}] // Flatten (* Jean-François Alcover, Dec 15 2017 *)
PROG
(PARI)
T(x, y) = x^y \\ Charles R Greathouse IV, Feb 07 2017
(SageMath)
def Arow(n, len): return [k**n for k in range(len)]
for n in range(8): print([n], Arow(n, 9)) # Peter Luschny, Apr 16 2024
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
New name by Peter Luschny, Apr 16 2024.
STATUS
approved