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A258993
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Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.
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10
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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, 66, 13, 1, 36, 210, 462, 495, 286, 91, 15, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1, 66, 715, 3003, 6435, 8008, 6188, 3060, 969, 190, 21
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OFFSET
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1,3
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COMMENTS
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rounded(T(n,k)/(2*k+1)) = A258708(n,k);
rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).
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LINKS
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FORMULA
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rounded(T(n,k)/(2*k+1)) = A258708(n,k);
rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).
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EXAMPLE
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. n\k | 0 1 2 3 4 5 6 7 8 9 10 11
. -----+-----------------------------------------------------------
. 1 | 1
. 2 | 1 3
. 3 | 1 6 5
. 4 | 1 10 15 7
. 5 | 1 15 35 28 9
. 6 | 1 21 70 84 45 11
. 7 | 1 28 126 210 165 66 13
. 8 | 1 36 210 462 495 286 91 15
. 9 | 1 45 330 924 1287 1001 455 120 17
. 10 | 1 55 495 1716 3003 3003 1820 680 153 19
. 11 | 1 66 715 3003 6435 8008 6188 3060 969 190 21
. 12 | 1 78 1001 5005 12870 19448 18564 11628 4845 1330 231 23 .
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MATHEMATICA
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Table[Binomial[n+k, n-k], {n, 1, 12}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Aug 01 2019 *)
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PROG
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(Haskell)
a258993 n k = a258993_tabl !! (n-1) !! k
a258993_row n = a258993_tabl !! (n-1)
a258993_tabl = zipWith (zipWith a007318) a094727_tabl a004736_tabl
(PARI) T(n, k) = binomial(n+k, n-k);
for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 01 2019
(Magma) [Binomial(n+k, n-k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Aug 01 2019
(Sage) [[binomial(n+k, n-k) for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Aug 01 2019
(GAP) Flat(List([1..12], n-> List([0..n-1], k-> Binomial(n+k, n-k) ))); # G. C. Greubel, Aug 01 2019
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CROSSREFS
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If a diagonal of 1's is added on the right, this becomes A085478.
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).
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).
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KEYWORD
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AUTHOR
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STATUS
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approved
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