A051128 Table T(n,k) = n^k read by upwards antidiagonals (n >= 1, k >= 1).

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 27, 16, 1, 6, 25, 64, 81, 32, 1, 7, 36, 125, 256, 243, 64, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 11, 100, 729, 4096, 16807, 46656, 78125, 65536, 19683, 1024, 1

Offset

1,2

Comments

Sum of antidiagonals is A003101(n) for n>0. - Alford Arnold, Jan 14 2007

Links

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

G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.

Formula

a(n) = A004736(n)^A002260(n) or ((t*t+3*t+4)/2-n)^(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012

Example

Table begins
1,    1,    1,    1,    1, ...
2,    4,    8,   16,   32, ...
3,    9,   27,   81,  243, ...
4,   16,   64,  256, 1024, ...

Maple

A051128 := proc(n)       # Boris Putievskiy's formula
a := floor((sqrt(8*n-7)+1)/2);
b := (a+a^2)/2-n;
c := (a-a^2)/2+n;
(b+1)^c end:
seq(A051128(n), n=1..61);  # Peter Luschny, Dec 14 2012
# second Maple program:
T:= (n, k)-> n^k:
seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # Alois P. Heinz, Apr 18 2020

Mathematica

Table[n^(k - n + 1), {k, 1, 11}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Dec 14 2012 *)

Prog

(PARI) T(n, k) = n^k \\ Charles R Greathouse IV, Feb 09 2017

Crossrefs

Cf. A051129, A003992, A004248, A033918, A095891, A220415, A220416, A220417, A003101.

Sequence in context: A284873 A107616 A055208 * A137614 A204213 A143326

Adjacent sequences: A051125 A051126 A051127 * A051129 A051130 A051131

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Dec 11 1999

Status

approved