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

Author

Roger L. Bagula, Feb 20 2009

Keywords

• nonn
• easy
• tabf

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

Examples

			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;
		

References

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

Links

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

Crossrefs

Cf. A132460, A113279, A082985, A029635.

Programs

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

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 *)

PARI

a(n,k)=n*binomial(n-k-1,k-1)/k; \\ Charles R Greathouse IV, Jul 10 2011

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

Formula

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

Extensions

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