A075363 Triangle read by rows, in which n-th row gives n smallest powers of n.

1, 2, 4, 3, 9, 27, 4, 16, 64, 256, 5, 25, 125, 625, 3125, 6, 36, 216, 1296, 7776, 46656, 7, 49, 343, 2401, 16807, 117649, 823543, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000

Offset: 1

Author

Amarnath Murthy, Sep 20 2002

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			From _Felix Fröhlich_, Sep 15 2019: (Start)
Triangle begins:
   1;
   2,   4;
   3,   9,   27;
   4,  16,   64,   256;
   5,  25,  125,   625,   3125;
   6,  36,  216,  1296,   7776,   46656;
   7,  49,  343,  2401,  16807,  117649,  823543;
   8,  64,  512,  4096,  32768,  262144, 2097152, 16777216;
   9,  81,  729,  6561,  59049,  531441, 4782969, 43046721, 387420489; (End)
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

Crossrefs

Cf. A075364, A068424.

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

Programs

Mathematica

Array[#^Range[#] &, 10] (* Paolo Xausa, Jun 09 2025 *)

PARI

row(n) = for(k=1, n, print1(n^k, ", "))
trianglerows(n) = for(x=1, n, row(x); print(""))
/* Print initial 10 rows as follows: */
trianglerows(10) \\ Felix Fröhlich, Sep 15 2019

Python

from math import isqrt, comb
def A075363(n): return (isqrt(n<<3)+1>>1)**(n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2)) # Chai Wah Wu, Jun 08 2025

Formula

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

a(n) = A002024(n)^A002260(n). [Gerald Hillier, Feb 12 2009]

Extensions

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

More terms from Michel Marcus, Sep 15 2019