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A258993 Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1. 10

%I

%S 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 66,13,1,36,210,462,495,286,91,15,1,45,330,924,1287,1001,455,120,17,1,

%U 55,495,1716,3003,3003,1820,680,153,19,1,66,715,3003,6435,8008,6188,3060,969,190,21

%N Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.

%C T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;

%C rounded(T(n,k)/(2*k+1)) = A258708(n,k);

%C rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).

%H Reinhard Zumkeller, <a href="/A258993/b258993.txt">Rows n = 1..125 of triangle, flattened</a>

%e . n\k | 0 1 2 3 4 5 6 7 8 9 10 11

%e . -----+-----------------------------------------------------------

%e . 1 | 1

%e . 2 | 1 3

%e . 3 | 1 6 5

%e . 4 | 1 10 15 7

%e . 5 | 1 15 35 28 9

%e . 6 | 1 21 70 84 45 11

%e . 7 | 1 28 126 210 165 66 13

%e . 8 | 1 36 210 462 495 286 91 15

%e . 9 | 1 45 330 924 1287 1001 455 120 17

%e . 10 | 1 55 495 1716 3003 3003 1820 680 153 19

%e . 11 | 1 66 715 3003 6435 8008 6188 3060 969 190 21

%e . 12 | 1 78 1001 5005 12870 19448 18564 11628 4845 1330 231 23 .

%o (Haskell)

%o a258993 n k = a258993_tabl !! (n-1) !! k

%o a258993_row n = a258993_tabl !! (n-1)

%o a258993_tabl = zipWith (zipWith a007318) a094727_tabl a004736_tabl

%Y If a diagonal of 1's is added on the right, this becomes A085478.

%Y Cf. A007318, A004736, A094727.

%Y Cf. A027941 (row sums), A117671 (central terms), A143858, A000967, A258708.

%Y 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 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 Cf. A165253.

%K nonn,tabl

%O 1,3

%A _Reinhard Zumkeller_, Jun 22 2015

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Last modified November 15 04:00 EST 2018. Contains 317225 sequences. (Running on oeis4.)