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A382253 Triangle T(n,k) = denominator of (n+k)/(1+n*k), 0 <= k <= n >= 0, read by rows.

%I A382253 #7 Apr 16 2025 10:25:24
%S A382253 1,1,1,1,1,5,1,1,7,5,1,1,3,13,17,1,1,11,2,7,13,1,1,13,19,5,31,37,1,1,
%T A382253 5,11,29,3,43,25,1,1,17,25,11,41,7,19,65,1,1,19,7,37,23,11,4,73,41,1,
%U A382253 1,7,31,41,17,61,71,9,91,101,1
%N A382253 Triangle T(n,k) = denominator of (n+k)/(1+n*k), 0 <= k <= n >= 0, read by rows.
%C A382253 Since the operation n @ k := (n + k)/(1 + n*k) is commutative, it is sufficient to list only the lower half of the "multiplication table", which would otherwise be an infinite square array. This triangle lists the denominators, and A382252 lists the numerators.
%F A382253 T(n,k) = T(k,n) for all n, k >= 0;
%F A382253 T(n,0) = T(0,n) = T(n,1) = T(1,n) = 1 for all n >= 0;
%F A382253 T(n,n) = denominator(2*n/(1+n^2)) = numerator((1+n^2)/2) = A228564(n).
%e A382253 The table for the operation n @ k := (n + k)/(1 + n*k) starts as follows:
%e A382253 (0 is the neutral element for the operation: n @ 0 = n = 0 @ n, therefore the elements in row 0 and column 0 equal the column and row index.)
%e A382253   0    1    2     3     4     5     6     7     8    Denominators of lower left
%e A382253   1    1    1     1     1     1     1     1     1    triangle:  1;
%e A382253   2    1   4/5   5/7   2/3   7/11  8/13  3/5  10/17             1, 1
%e A382253   3    1   5/7   3/5   7/13  1/2   9/19  5/11 11/25             1, 1, 5;
%e A382253   4    1   2/3   7/13  8/17  3/7   2/5  11/29  4/11             1, 1, 7, 5;
%e A382253   5    1   7/11  1/2   3/7   5/13 11/31  1/3  13/41             1, 1, 3, 13, 17;
%e A382253   6    1   8/13  9/19  2/5  11/31 12/37 13/43  2/7                 etc.
%e A382253   7    1   3/5   5/11 11/29  1/3  13/43  7/25  5/19
%e A382253   8    1  10/17 11/25  4/11 13/41  2/7   5/19 16/65
%e A382253 The sequence lists the denominators of the values, where denominator(x) = 1 for integers, and only for the lower left triangle of the table, by rows.
%o A382253 (PARI) apply( {A382253(n,k=-1)= k<0&& k=n-(1+n=(sqrtint(8*n+1)-1)\2)*n/2; denominator((n+k)/(1+n*k))}, [0..66])
%Y A382253 Cf. A382252, A382257; A228564 (main diagonal), A001477 (row & col. 0 of the '@' table), A000012 (row & col. 0 & 1 of the table of denominators).
%K A382253 nonn,tabl,frac
%O A382253 0,6
%A A382253 _M. F. Hasler_, Apr 15 2025