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A385850 Triangle read by rows: T(n,k) = denominator((Sum_{i=1..k} (n-i+1)^2)/(Sum_{i=1..k} i^2)), with 1 <= k <= n.

%I A385850 #9 Jul 14 2025 22:09:06
%S A385850 1,1,1,1,5,1,1,1,14,1,1,5,7,5,1,1,5,2,15,11,1,1,1,7,5,11,91,1,1,5,14,
%T A385850 5,11,91,20,1,1,1,7,3,11,91,1,51,1,1,5,2,5,1,91,20,51,95,1,1,5,7,5,11,
%U A385850 91,5,17,95,77,1,1,1,14,15,11,7,4,51,95,77,46,1
%N A385850 Triangle read by rows: T(n,k) = denominator((Sum_{i=1..k} (n-i+1)^2)/(Sum_{i=1..k} i^2)), with 1 <= k <= n.
%H A385850 Stefano Spezia, <a href="/A385850/b385850.txt">Table of n, a(n) for n = 1..11325</a> (first 150 rows of the triangle, flattened)
%F A385850 T(n,k) = denominator((1 - 3*k + 2*k^2 + 6*n - 6*k*n + 6*n^2)/(1 + 3*k + 2*k^2)).
%e A385850 Triangle of the fractions begins as:
%e A385850    1/1;
%e A385850    4/1,  1/1;
%e A385850    9/1, 13/5,    1/1;
%e A385850   16/1,  5/1,  29/14,    1/1;
%e A385850   25/1, 41/5,   25/7,    9/5,    1/1;
%e A385850   36/1, 61/5,   11/2,  43/15,  18/11,     1/1;
%e A385850   49/1, 17/1,   55/7,   21/5,  27/11,  139/91,  1/1;
%e A385850   ...
%e A385850 A385849(4,3)/T(4,3) = (4^2 + 3^2 + 2^2)/(1^2 + 2^2 + 3^2) = 29/14.
%t A385850 T[n_,k_]:=Denominator[(1-3k+2k^2+6n-6k*n+6n^2)/(1+3k+2k^2)]; Table[T[n,k],{n,12},{k,n}]//Flatten
%Y A385850 Cf. A000012 (1st column or diagonal), A385849 (numerators).
%K A385850 nonn,easy,frac,look,tabl
%O A385850 1,5
%A A385850 _Stefano Spezia_, Jul 10 2025