A220415 -id:A220415 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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.

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

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

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

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

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

More terms from James Sellers, Dec 11 1999

A033918 Triangular array in which n-th row consists of the numbers 1^1, 2^2, ... n^n.

Original entry on oeis.org

1, 1, 4, 1, 4, 27, 1, 4, 27, 256, 1, 4, 27, 256, 3125, 1, 4, 27, 256, 3125, 46656, 1, 4, 27, 256, 3125, 46656, 823543, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 1, 4, 27, 256, 3125, 46656

Offset: 1

Timur I Khantimirov (Tim(AT)sbbank.udm.ru)

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A033918 is the reluctant sequence of A000312 (number of labeled mappings from n points to themselves, endofunctions): n^n. - Boris Putievskiy, Dec 14 2012

			1;
1, 4;
1, 4, 27;
1, 4, 27, 256;
1, 4, 27, 256, 3125;
1, 4, 27, 256, 3125, 46656;
1, 4, 27, 256, 3125, 46656, 823543;
...

Timur I Khantimirov and Boris Putievskiy (first 51 from Timur I Khantimirov), Table of n, a(n) for n = 1..1000
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002260, A220415, A220416.

Module[{nn=10,c},c=Table[n^n,{n,nn}];Flatten[Table[Take[c,i],{i,nn}]]] (* Harvey P. Dale, Nov 02 2014 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
m=(n-t*(t+1)/2)**(n-t*(t+1)/2)

a(n) = A000312(m), where m= n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2) or a(n)=(n-t(t+1)/2)^(n-t(t+1)/2), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012

cp's OEIS Frontend

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

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