A090414 -id:A090414 - cp's OEIS frontend

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

Amarnath Murthy, Sep 20 2002

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

			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)

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

Cf. A075364, A068424.

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

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

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

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

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

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

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

More terms from Michel Marcus, Sep 15 2019

cp's OEIS Frontend

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

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