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A213999 Denominators of the triangle of fractions read by rows: pf(n,0) = 1, pf(n,n) = 1/(n+1) and pf(n+1,k) = pf(n,k) + pf(n,k-1) with 0 < k < n; denominators: A213998.

%I A213999 #15 Nov 10 2021 07:30:20
%S A213999 1,1,2,1,2,3,1,2,6,4,1,2,3,12,5,1,2,6,12,60,6,1,2,3,4,10,20,7,1,2,6,
%T A213999 12,20,20,140,8,1,2,3,12,15,10,35,280,9,1,2,6,4,20,30,70,280,2520,10,
%U A213999 1,2,3,12,10,12,21,56,252,2520,11,1,2,6,12,60,60,84,168,504,2520,27720,12
%N A213999 Denominators of the triangle of fractions read by rows: pf(n,0) = 1, pf(n,n) = 1/(n+1) and pf(n+1,k) = pf(n,k) + pf(n,k-1) with 0 < k < n; denominators: A213998.
%C A213999 T(n,0) = 1;
%C A213999 T(n,1) = A007395(n) for n > 0;
%C A213999 T(n,2) = A010704(n) for n > 1;
%C A213999 T(n,n-3) = A124838(n-2) for n > 2;
%C A213999 T(n,n-2) = A027611(n-1) for n > 1;
%C A213999 T(n,n-1) = A002805(n) for n > 0;
%C A213999 T(n,n) = n + 1;
%C A213999 A003418(n+1) = least common multiple of n-th row;
%C A213999 A214075(n,k) = floor(A213998(n,k) / T(n,k)).
%H A213999 Reinhard Zumkeller, <a href="/A213999/b213999.txt">Rows n = 0..150 of triangle, flattened</a>
%H A213999 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%e A213999 See A213998.
%t A213999 T[_, 0] = 1; T[n_, n_] := 1/(n + 1);
%t A213999 T[n_, k_] := T[n, k] = T[n - 1, k] + T[n - 1, k - 1];
%t A213999 Table[T[n, k] // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 10 2021 *)
%o A213999 (Haskell)
%o A213999 import Data.Ratio ((%), denominator, Ratio)
%o A213999 a213999 n k = a213999_tabl !! n !! k
%o A213999 a213999_row n = a213999_tabl !! n
%o A213999 a213999_tabl = map (map denominator) $ iterate pf [1] where
%o A213999    pf row = zipWith (+) ([0] ++ row) (row ++ [-1 % (x * (x + 1))])
%o A213999             where x = denominator $ last row
%K A213999 nonn,frac,tabl,look
%O A213999 0,3
%A A213999 _Reinhard Zumkeller_, Jul 03 2012