%I #20 Jul 28 2023 18:27:43

%S 1,2,4,3,9,27,4,16,64,256,5,25,125,625,3125,6,36,216,1296,7776,46656,

%T 7,49,343,2401,16807,117649,823543,8,64,512,4096,32768,262144,2097152,

%U 16777216,9,81,729,6561,59049,531441,4782969,43046721,387420489,10,100,1000,10000,100000,1000000,10000000,100000000,1000000000,10000000000

%N Triangle read by rows, in which n-th row gives n smallest powers of n.

%C T(n,k) is the number of sequences with repetition (k-tuples) of k (not necessarily different) elements taken from an n-set S. These sequences are also called "words of length k over the alphabet S". For sequences without repetition (partial permutations) cf. A068424. - _Manfred Boergens_, Jun 18 2023

%F T(n, k) = n^k, 1<=k<=n.

%F a(n) = A002024(n)^A002260(n). [_Gerald Hillier_, Feb 12 2009]

%e From _Felix Fröhlich_, Sep 15 2019: (Start)

%e Triangle begins:

%e 1;

%e 2, 4;

%e 3, 9, 27;

%e 4, 16, 64, 256;

%e 5, 25, 125, 625, 3125;

%e 6, 36, 216, 1296, 7776, 46656;

%e 7, 49, 343, 2401, 16807, 117649, 823543;

%e 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216;

%e 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489; (End)

%o (PARI) row(n) = for(k=1, n, print1(n^k, ", "))

%o trianglerows(n) = for(x=1, n, row(x); print(""))

%o /* Print initial 10 rows as follows: */

%o trianglerows(10) \\ _Felix Fröhlich_, Sep 15 2019

%Y Cf. A075364, A068424.

%Y T(n, 1) = A000027(n), T(n, n) = A000312(n). Cf. A090414.

%K nonn,tabl,easy

%O 1,2

%A _Amarnath Murthy_, Sep 20 2002

%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

%E More terms from _Michel Marcus_, Sep 15 2019