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A157000
Triangle T(n,k) = (n/k)*binomial(n-k-1, k-1) read by rows.
4
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
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
(SageMath) [[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
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