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A076112 Triangle (read by rows) in which the n-th row contains first n terms of n geometric progression with first term 1 and common ratio (n-1).

%I A076112 #27 Jun 09 2025 00:52:17
%S A076112 1,1,2,1,3,9,1,4,16,64,1,5,25,125,625,1,6,36,216,1296,7776,1,7,49,343,
%T A076112 2401,16807,117649,1,8,64,512,4096,32768,262144,2097152,1,9,81,729,
%U A076112 6561,59049,531441,4782969,43046721,1,10,100,1000,10000,100000,1000000
%N A076112 Triangle (read by rows) in which the n-th row contains first n terms of n geometric progression with first term 1 and common ratio (n-1).
%C A076112 Table T(n,k) = (n+k-1)^(n-1) n,k > 0 read by antidiagonals. - _Boris Putievskiy_, Dec 17 2012
%H A076112 Harvey P. Dale, <a href="/A076112/b076112.txt">Table of n, a(n) for n = 1..5000</a>
%H A076112 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A076112 As a linear array, the sequence is a(n) = A002024(n)^A002260(n) or a(n) = (t+1)^(n-t*(t+1)/2-1), where t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 17 2012
%e A076112 Triangle begins:
%e A076112   1;
%e A076112   1,2;
%e A076112   1,3,9;
%e A076112   1,4,16,64;
%e A076112   1,5,25,125,625;
%e A076112   1,6,36,216,1296,7776;
%e A076112   ...
%t A076112 Table[n^Range[0,n-1],{n,10}]//Flatten (* _Harvey P. Dale_, Jan 27 2020 *)
%o A076112 (Python)
%o A076112 from math import isqrt, comb
%o A076112 def A076112(n): return (isqrt(n<<3)+1>>1)**(n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2)-1) # _Chai Wah Wu_, Jun 09 2025
%Y A076112 Cf. A000169, A076113.
%K A076112 easy,nonn,tabl
%O A076112 1,3
%A A076112 _Amarnath Murthy_, Oct 09 2002
%E A076112 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003