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

1,2

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

Links

Table of n, a(n) for n=1..55.

Formula

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

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

Example

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)

Prog

(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

Crossrefs

Cf. A075364, A068424.

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

Sequence in context: A110339 A223703 A157406 * A082382 A239599 A271864

Adjacent sequences: A075360 A075361 A075362 * A075364 A075365 A075366

Keyword

nonn,tabl,easy

Author

Amarnath Murthy, Sep 20 2002

Extensions

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

More terms from Michel Marcus, Sep 15 2019

Status

approved