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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.
11

%I #15 Nov 10 2021 07:30:20

%S 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 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 1,2,3,12,10,12,21,56,252,2520,11,1,2,6,12,60,60,84,168,504,2520,27720,12

%N 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 T(n,0) = 1;

%C T(n,1) = A007395(n) for n > 0;

%C T(n,2) = A010704(n) for n > 1;

%C T(n,n-3) = A124838(n-2) for n > 2;

%C T(n,n-2) = A027611(n-1) for n > 1;

%C T(n,n-1) = A002805(n) for n > 0;

%C T(n,n) = n + 1;

%C A003418(n+1) = least common multiple of n-th row;

%C A214075(n,k) = floor(A213998(n,k) / T(n,k)).

%H Reinhard Zumkeller, <a href="/A213999/b213999.txt">Rows n = 0..150 of triangle, flattened</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%e See A213998.

%t T[_, 0] = 1; T[n_, n_] := 1/(n + 1);

%t T[n_, k_] := T[n, k] = T[n - 1, k] + T[n - 1, k - 1];

%t Table[T[n, k] // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 10 2021 *)

%o (Haskell)

%o import Data.Ratio ((%), denominator, Ratio)

%o a213999 n k = a213999_tabl !! n !! k

%o a213999_row n = a213999_tabl !! n

%o a213999_tabl = map (map denominator) $ iterate pf [1] where

%o pf row = zipWith (+) ([0] ++ row) (row ++ [-1 % (x * (x + 1))])

%o where x = denominator $ last row

%K nonn,frac,tabl,look

%O 0,3

%A _Reinhard Zumkeller_, Jul 03 2012