This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A213998 #21 Nov 10 2021 07:33:30
%S A213998 1,1,1,1,3,1,1,5,11,1,1,7,13,25,1,1,9,47,77,137,1,1,11,37,57,87,49,1,
%T A213998 1,13,107,319,459,223,363,1,1,15,73,533,743,341,481,761,1,1,17,191,
%U A213998 275,1879,2509,3349,4609,7129,1,1,19,121,1207,1627,2131,2761,3601,4861,7381,1
%N A213998 Numerators 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.
%C A213998 T(n,0) = 1;
%C A213998 T(n,1) = A005408(n-1) for n > 0;
%C A213998 T(n,2) = A188386(n-2) for n > 2;
%C A213998 T(n,n-3) = A124837(n-2) for n > 2;
%C A213998 T(n,n-2) = A027612(n-1) for n > 1;
%C A213998 T(n,n-1) = A001008(n) for n > 0;
%C A213998 T(n,n) = 1;
%C A213998 A214075(n,k) = floor(T(n,k) / A213999(n,k)).
%H A213998 Reinhard Zumkeller, <a href="/A213998/b213998.txt">Rows n = 0..150 of triangle, flattened</a>
%H A213998 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%e A213998 Start of triangle pf with corresponding triangles of numerators and denominators:
%e A213998 . 0: 1
%e A213998 . 1: 1 1/2
%e A213998 . 2: 1 3/2 1/3
%e A213998 . 3: 1 5/2 11/6 1/4
%e A213998 . 4: 1 7/2 13/3 25/12 1/5
%e A213998 . 5: 1 9/2 47/6 77/12 137/60 1/6
%e A213998 . 6: 1 11/2 37/3 57/4 87/10 49/20 1/7
%e A213998 . 7: 1 13/2 107/6 319/12 459/20 223/20 363/140 1/8
%e A213998 . 8: 1 15/2 73/3 533/12 743/15 341/10 481/35 761/280 1/9,
%e A213998 .
%e A213998 . 0: numerators 1 1 denominators
%e A213998 . 1: 1 1 1 2 A213999
%e A213998 . 2: 1 3 1 1 2 3
%e A213998 . 3: 1 5 11 1 1 2 6 4
%e A213998 . 4: 1 7 13 25 1 1 2 3 12 5
%e A213998 . 5: 1 9 47 77 137 1 1 2 6 12 60 6
%e A213998 . 6: 1 11 37 57 87 49 1 1 2 3 4 10 20 7
%e A213998 . 7: 1 13 107 319 459 223 363 1 1 2 6 12 20 20 140 8
%e A213998 . 8: 1 15 73 533 743 341 481 761 1, 1 2 3 12 15 10 35 280 9.
%t A213998 T[_, 0] = 1; T[n_, n_] := 1/(n + 1);
%t A213998 T[n_, k_] := T[n, k] = T[n - 1, k] + T[n - 1, k - 1];
%t A213998 Table[T[n, k] // Numerator, {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 10 2021 *)
%o A213998 (Haskell)
%o A213998 import Data.Ratio ((%), numerator, denominator, Ratio)
%o A213998 a213998 n k = a213998_tabl !! n !! k
%o A213998 a213998_row n = a213998_tabl !! n
%o A213998 a213998_tabl = map (map numerator) $ iterate pf [1] where
%o A213998 pf row = zipWith (+) ([0] ++ row) (row ++ [-1 % (x * (x + 1))])
%o A213998 where x = denominator $ last row
%Y A213998 Cf. A005408, A188386 (columns).
%Y A213998 Cf. A001008, A027612, A124837 (diagonals).
%Y A213998 Cf. A213999 (denominators).
%K A213998 nonn,frac,tabl,look
%O A213998 0,5
%A A213998 _Reinhard Zumkeller_, Jul 03 2012