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A377188 Triangle read by rows: T(n,k) = denominator((n^2 + k)/(n^2 - k)).

%I A377188 #11 Nov 13 2024 01:55:11
%S A377188 1,7,1,7,13,3,23,11,21,2,17,11,4,31,5,47,23,45,22,43,3,31,61,15,59,29,
%T A377188 57,7,79,13,77,38,25,37,73,4,49,97,12,19,47,93,23,91,9,119,59,117,58,
%U A377188 115,57,113,56,111,5,71,47,35,139,23,137,17,15,67,133,11
%N A377188 Triangle read by rows: T(n,k) = denominator((n^2 + k)/(n^2 - k)).
%D A377188 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.3, p. 14.
%F A377188 Limit_{n->oo} Product_{k=1..n} A377187(n,k)/T(n,k) = e = A001113 (see Finch).
%F A377188 T(n,n) = A026741(n-1).
%e A377188 The triangle begins as:
%e A377188    1;
%e A377188    7,  1;
%e A377188    7, 13,  3;
%e A377188   23, 11, 21,  2;
%e A377188   17, 11,  4, 31,  5;
%e A377188   47, 23, 45, 22, 43,  3;
%e A377188   31, 61, 15, 59, 29, 57,  7;
%e A377188   79, 13, 77, 38, 25, 37, 73,  4;
%e A377188   49, 97, 12, 19, 47, 93, 23, 91, 9;
%e A377188   ...
%t A377188 T[n_,k_]:=Denominator[(n^2+k)/(n^2-k)]; Table[T[n,k],{n,2,12},{k,2,n}]//Flatten
%o A377188 (Python)
%o A377188 from math import isqrt, comb, gcd
%o A377188 def A377188(n): return (d:=(a:=(m:=isqrt(k:=n-1<<1))+(k>m*(m+1))+1)**2-(b:=n-comb(a-1,2)))//gcd(d,d+(b<<1)) # _Chai Wah Wu_, Nov 12 2024
%Y A377188 Cf. A001113, A026741, A377187 (numerator).
%K A377188 nonn,easy,frac,tabl
%O A377188 2,2
%A A377188 _Stefano Spezia_, Oct 19 2024