cp's OEIS Frontend

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

%I A075363 #28 Jun 09 2025 10:40:23
%S A075363 1,2,4,3,9,27,4,16,64,256,5,25,125,625,3125,6,36,216,1296,7776,46656,
%T A075363 7,49,343,2401,16807,117649,823543,8,64,512,4096,32768,262144,2097152,
%U A075363 16777216,9,81,729,6561,59049,531441,4782969,43046721,387420489,10,100,1000,10000,100000,1000000,10000000,100000000,1000000000,10000000000
%N A075363 Triangle read by rows, in which n-th row gives n smallest powers of n.
%C A075363 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
%H A075363 Paolo Xausa, <a href="/A075363/b075363.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of triangle, flattened).
%F A075363 T(n, k) = n^k, 1<=k<=n.
%F A075363 a(n) = A002024(n)^A002260(n). [_Gerald Hillier_, Feb 12 2009]
%e A075363 From _Felix Fröhlich_, Sep 15 2019: (Start)
%e A075363 Triangle begins:
%e A075363    1;
%e A075363    2,   4;
%e A075363    3,   9,   27;
%e A075363    4,  16,   64,   256;
%e A075363    5,  25,  125,   625,   3125;
%e A075363    6,  36,  216,  1296,   7776,   46656;
%e A075363    7,  49,  343,  2401,  16807,  117649,  823543;
%e A075363    8,  64,  512,  4096,  32768,  262144, 2097152, 16777216;
%e A075363    9,  81,  729,  6561,  59049,  531441, 4782969, 43046721, 387420489; (End)
%t A075363 Array[#^Range[#] &, 10] (* _Paolo Xausa_, Jun 09 2025 *)
%o A075363 (PARI) row(n) = for(k=1, n, print1(n^k, ", "))
%o A075363 trianglerows(n) = for(x=1, n, row(x); print(""))
%o A075363 /* Print initial 10 rows as follows: */
%o A075363 trianglerows(10) \\ _Felix Fröhlich_, Sep 15 2019
%o A075363 (Python)
%o A075363 from math import isqrt, comb
%o A075363 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
%Y A075363 Cf. A075364, A068424.
%Y A075363 T(n, 1) = A000027(n), T(n, n) = A000312(n). Cf. A090414.
%K A075363 nonn,tabl,easy
%O A075363 1,2
%A A075363 _Amarnath Murthy_, Sep 20 2002
%E A075363 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
%E A075363 More terms from _Michel Marcus_, Sep 15 2019