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 A258993 #21 Nov 16 2023 05:18:45
%S A258993 1,1,3,1,6,5,1,10,15,7,1,15,35,28,9,1,21,70,84,45,11,1,28,126,210,165,
%T A258993 66,13,1,36,210,462,495,286,91,15,1,45,330,924,1287,1001,455,120,17,1,
%U A258993 55,495,1716,3003,3003,1820,680,153,19,1,66,715,3003,6435,8008,6188,3060,969,190,21
%N A258993 Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.
%C A258993 T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;
%C A258993 rounded(T(n,k)/(2*k+1)) = A258708(n,k);
%C A258993 rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).
%H A258993 Reinhard Zumkeller, <a href="/A258993/b258993.txt">Rows n = 1..125 of triangle, flattened</a>
%F A258993 T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;
%F A258993 rounded(T(n,k)/(2*k+1)) = A258708(n,k);
%F A258993 rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).
%e A258993 . n\k | 0 1 2 3 4 5 6 7 8 9 10 11
%e A258993 . -----+-----------------------------------------------------------
%e A258993 . 1 | 1
%e A258993 . 2 | 1 3
%e A258993 . 3 | 1 6 5
%e A258993 . 4 | 1 10 15 7
%e A258993 . 5 | 1 15 35 28 9
%e A258993 . 6 | 1 21 70 84 45 11
%e A258993 . 7 | 1 28 126 210 165 66 13
%e A258993 . 8 | 1 36 210 462 495 286 91 15
%e A258993 . 9 | 1 45 330 924 1287 1001 455 120 17
%e A258993 . 10 | 1 55 495 1716 3003 3003 1820 680 153 19
%e A258993 . 11 | 1 66 715 3003 6435 8008 6188 3060 969 190 21
%e A258993 . 12 | 1 78 1001 5005 12870 19448 18564 11628 4845 1330 231 23 .
%t A258993 Table[Binomial[n+k,n-k], {n,1,12}, {k,0,n-1}]//Flatten (* _G. C. Greubel_, Aug 01 2019 *)
%o A258993 (Haskell)
%o A258993 a258993 n k = a258993_tabl !! (n-1) !! k
%o A258993 a258993_row n = a258993_tabl !! (n-1)
%o A258993 a258993_tabl = zipWith (zipWith a007318) a094727_tabl a004736_tabl
%o A258993 (PARI) T(n,k) = binomial(n+k,n-k);
%o A258993 for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Aug 01 2019
%o A258993 (Magma) [Binomial(n+k,n-k): k in [0..n-1], n in [1..12]]; // _G. C. Greubel_, Aug 01 2019
%o A258993 (Sage) [[binomial(n+k,n-k) for k in (0..n-1)] for n in (1..12)] # _G. C. Greubel_, Aug 01 2019
%o A258993 (GAP) Flat(List([1..12], n-> List([0..n-1], k-> Binomial(n+k,n-k) ))); # _G. C. Greubel_, Aug 01 2019
%Y A258993 If a diagonal of 1's is added on the right, this becomes A085478.
%Y A258993 Essentially the same as A143858.
%Y A258993 Cf. A007318, A004736, A094727.
%Y A258993 Cf. A027941 (row sums), A117671 (central terms), A143858, A000967, A258708.
%Y A258993 T(n,k): A000217 (k=1), A000332 (k=2), A000579 (k=3), A000581 (k=4), A001287 (k=5), A010965 (k=6), A010967 (k=7), A010969 (k=8), A010971 (k=9), A010973 (k=10), A010975 (k=11), A010977 (k=12), A010979 (k=13), A010981 (k=14), A010983 (k=15), A010985 (k=16), A010987 (k=17), A010989 (k=18), A010991 (k=19), A010993 (k=20), A010995 (k=21), A010997 (k=22), A010999 (k=23), A011001 (k=24), A017714 (k=25), A017716 (k=26), A017718 (k=27), A017720 (k=28), A017722 (k=29), A017724 (k=30), A017726 (k=31), A017728 (k=32), A017730 (k=33), A017732 (k=34), A017734 (k=35), A017736 (k=36), A017738 (k=37), A017740 (k=38), A017742 (k=39), A017744 (k=40), A017746 (k=41), A017748 (k=42), A017750 (k=43), A017752 (k=44), A017754 (k=45), A017756 (k=46), A017758 (k=47), A017760 (k=48), A017762 (k=49), A017764 (k=50).
%Y A258993 T(n+k,n): A005408 (k=1), A000384 (k=2), A000447 (k=3), A053134 (k=4), A002299 (k=5), A053135 (k=6), A053136 (k=7), A053137 (k=8), A053138 (k=9), A196789 (k=10).
%Y A258993 Cf. A165253.
%K A258993 nonn,tabl
%O A258993 1,3
%A A258993 _Reinhard Zumkeller_, Jun 22 2015