A258993 - OEIS

A258993

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

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

1,3

COMMENTS

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

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

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

LINKS

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

FORMULA

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

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

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

EXAMPLE

. 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 .

MATHEMATICA

Table[Binomial[n+k, n-k], {n, 1, 12}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

PROG

(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

CROSSREFS

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

Essentially the same as A143858.

Cf. A007318, A004736, A094727.

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

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).

Cf. A165253.

Sequence in context: A061702 A112351 A143858 * A109954 A355010 A153641

Adjacent sequences: A258990 A258991 A258992 * A258994 A258995 A258996

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, Jun 22 2015

STATUS

approved