A157000 Triangle T(n,k) = (n/k)*binomial(n-k-1, k-1) read by rows.

2, 3, 4, 2, 5, 5, 6, 9, 2, 7, 14, 7, 8, 20, 16, 2, 9, 27, 30, 9, 10, 35, 50, 25, 2, 11, 44, 77, 55, 11, 12, 54, 112, 105, 36, 2, 13, 65, 156, 182, 91, 13, 14, 77, 210, 294, 196, 49, 2, 15, 90, 275, 450, 378, 140, 15, 16, 104, 352, 660, 672, 336, 64, 2, 17, 119, 442, 935, 1122, 714, 204, 17

Offset

2,1

Comments

Row sums are A001610(n-1).

Triangle A034807 (coefficients of Lucas polynomials) with the first column omitted. - Philippe Deléham, Mar 17 2013

T(n,k) is the number of ways to select k knights from a round table of n knights, no two adjacent. - Bert Seghers, Mar 02 2014

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 199

Links

G. C. Greubel, Rows n = 2..100 of triangle, flattened

Formula

T(n,k) = binomial(n-k,k) + binomial(n-k-1,k-1). - Bert Seghers, Mar 02 2014

Example

The table starts in row n=2, column k=1 as:
   2;
   3;
   4,  2;
   5,  5;
   6,  9,   2;
   7, 14,   7;
   8, 20,  16,   2;
   9, 27,  30,   9;
  10, 35,  50,  25,  2;
  11, 44,  77,  55, 11;
  12, 54, 112, 105, 36, 2;

Mathematica

Table[(n/k)*Binomial[n-k-1, k-1], {n, 2, 20}, {k, 1, Floor[n/2]}]//Flatten (* modified by G. C. Greubel, Apr 25 2019 *)

Prog

(PARI) a(n, k)=n*binomial(n-k-1, k-1)/k; \\ Charles R Greathouse IV, Jul 10 2011
(Magma) [[n*Binomial(n-k-1, k-1)/k: k in [1..Floor(n/2)]]: n in [2..20]]; // G. C. Greubel, Apr 25 2019
(Sage) [[n*binomial(n-k-1, k-1)/k for k in (1..floor(n/2))] for n in (2..20)] # G. C. Greubel, Apr 25 2019

Crossrefs

Cf. A132460, A113279, A082985, A029635.

Sequence in context: A300840 A243849 A286547 * A026346 A340791 A369422

Adjacent sequences: A156997 A156998 A156999 * A157001 A157002 A157003

Keyword

nonn,easy,tabf

Author

Roger L. Bagula, Feb 20 2009

Extensions

Offset 2, keyword:tabf, more terms by the Assoc. Eds. of the OEIS, Nov 01 2010

Status

approved