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A130296 Triangle read by rows: T[i,1]=i, T[i,j]=1 for 1 < j <= i = 1,2,3,...

%I A130296 #14 Nov 09 2024 11:37:29
%S A130296 1,2,1,3,1,1,4,1,1,1,5,1,1,1,1,6,1,1,1,1,1,7,1,1,1,1,1,1,8,1,1,1,1,1,
%T A130296 1,1,9,1,1,1,1,1,1,1,1,10,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,12,
%U A130296 1,1,1,1,1,1,1,1,1,1,1,13,1,1,1,1,1,1,1,1,1,1,1,1,14,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A130296 Triangle read by rows: T[i,1]=i, T[i,j]=1 for 1 < j <= i = 1,2,3,...
%C A130296 This sequence was formerly named "reversals of A051340", but it is actually the truncation of A051340 to its lower left triangular part, re-indexed to start rows and columns with 1. - _M. F. Hasler_, Aug 15 2015
%F A130296 Truncation of A051340 to its lower left. By rows, "n" followed by (n-1) 1's. (1,2,3...) in the left border, all 1's in other columns.
%F A130296 A130296^2 = A130297.
%F A130296 a(n) = A004201(n) - A004201(n-1) for n>1. - _Reinhard Zumkeller_, Jul 16 2008
%e A130296 First few rows of the triangle are:
%e A130296 1;
%e A130296 2, 1;
%e A130296 3, 1, 1;
%e A130296 4, 1, 1, 1;
%e A130296 5, 1, 1, 1, 1;
%e A130296 ...
%o A130296 (PARI) A130296(i,j)=if(j==1,i,j<=i) \\ The sequence should not be defined for j>i but it is used in several places as infinite square matrix with upper right part equal to zero. - _M. F. Hasler_, Aug 15 2015
%o A130296 (Python)
%o A130296 from math import isqrt
%o A130296 def A130296(n): return comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2)-comb((m2:=isqrt(k-2))+(k-2>m2*(m2+1)),2)+1 # _Chai Wah Wu_, Nov 09 2024
%Y A130296 Cf. A051340, A130297, A005408 (row sums).
%K A130296 nonn,tabl,easy
%O A130296 1,2
%A A130296 _Gary W. Adamson_, May 20 2007