A156003 Triangle read by rows: T(n, k) = binomial(3*n, k-1) + binomial(3*n, n-k).

2, 7, 7, 37, 18, 37, 221, 78, 78, 221, 1366, 470, 210, 470, 1366, 8569, 3078, 969, 969, 3078, 8569, 54265, 20370, 6195, 2660, 6195, 20370, 54265, 346105, 134620, 42780, 12650, 12650, 42780, 134620, 346105, 2220076, 888057, 296361, 83655, 35100, 83655, 296361, 888057, 2220076

Offset

1,1

Comments

Row sums are: {2, 14, 92, 598, 3882, 25232, 164320, 1072310, 7011398, 45928174, ...}.

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

B. Brainerd and T. V. Narayana, A Note on Simple Binomial Sampling Plans, Ann. Math. Statist. Volume 32, Number 3 (1961), 906-908.

Formula

T(n, k) = binomial(3*n, k-1) + binomial(3*n, n-k).

Example

Triangle begins as:
        2;
        7,      7;
       37,     18,     37;
      221,     78,     78,   221;
     1366,    470,    210,   470,  1366;
     8569,   3078,    969,   969,  3078,  8569;
    54265,  20370,   6195,  2660,  6195, 20370,  54265;
   346105, 134620,  42780, 12650, 12650, 42780, 134620, 346105;
  2220076, 888057, 296361, 83655, 35100, 83655, 296361, 888057, 2220076;

Maple

b:=binomial; seq(seq( b(3*n, k-1) + b(3*n, n-k), k=1..n), n=1..10); # G. C. Greubel, Dec 01 2019

Mathematica

Table[Binomial[3*n, k-1] + Binomial[3*n, n-k], {n, 10}, {k, n}]//Flatten

Prog

(PARI) T(n, k) = my(b=binomial); b(3*n, k-1) + b(3*n, n-k); \\ G. C. Greubel, Dec 01 2019
(Magma) B:=Binomial; [B(3*n, k-1) + B(3*n, n-k): k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 01 2019
(Sage) b=binomial; [[b(3*n, k-1) + b(3*n, n-k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Dec 01 2019
(GAP) B:=Binomial;; Flat(List([1..10], n-> List([1..n], k-> B(3*n, k-1) + B(3*n, n-k) ))); # G. C. Greubel, Dec 01 2019

Crossrefs

Sequence in context: A090521 A090523 A164314 * A306417 A011416 A086658

Adjacent sequences: A156000 A156001 A156002 * A156004 A156005 A156006

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 01 2009

Status

approved