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A385849 Triangle read by rows: T(n,k) = numerator((Sum_{i=1..k} (n-i+1)^2)/(Sum_{i=1..k} i^2)), with 1 <= k <= n.

%I A385849 #15 Jul 14 2025 22:09:02
%S A385849 1,4,1,9,13,1,16,5,29,1,25,41,25,9,1,36,61,11,43,18,1,49,17,55,21,27,
%T A385849 139,1,64,113,149,29,38,199,29,1,81,29,97,23,51,271,2,71,1,100,181,35,
%U A385849 49,6,355,53,95,128,1,121,221,151,61,83,451,17,41,167,101,1
%N A385849 Triangle read by rows: T(n,k) = numerator((Sum_{i=1..k} (n-i+1)^2)/(Sum_{i=1..k} i^2)), with 1 <= k <= n.
%H A385849 Stefano Spezia, <a href="/A385849/b385849.txt">Table of n, a(n) for n = 1..11325</a> (first 150 rows of the triangle, flattened)
%H A385849 Michael De Vlieger, <a href="/A385849/a385849.png">Scatterplot of a(n)</a>, n = 1..11325.
%F A385849 T(n,k) = numerator((1 - 3*k + 2*k^2 + 6*n - 6*k*n + 6*n^2)/(1 + 3*k + 2*k^2)).
%e A385849 Triangle of the fractions begins as:
%e A385849    1/1;
%e A385849    4/1,  1/1;
%e A385849    9/1, 13/5,    1/1;
%e A385849   16/1,  5/1,  29/14,    1/1;
%e A385849   25/1, 41/5,   25/7,    9/5,    1/1;
%e A385849   36/1, 61/5,   11/2,  43/15,  18/11,     1/1;
%e A385849   49/1, 17/1,   55/7,   21/5,  27/11,  139/91,  1/1;
%e A385849   ...
%e A385849 T(4,3)/A385850(4,3) = (4^2 + 3^2 + 2^2)/(1^2 + 2^2 + 3^2) = 29/14.
%t A385849 T[n_,k_]:=Numerator[(1-3k+2k^2+6n-6k*n+6n^2)/(1+3k+2k^2)]; Table[T[n,k],{n,11},{k,n}]//Flatten
%Y A385849 Cf. A000012 (diagonal), A000290 (1st column), A322135, A385850 (denominators).
%K A385849 nonn,easy,frac,look,tabl
%O A385849 1,2
%A A385849 _Stefano Spezia_, Jul 10 2025