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

%I #25 Sep 08 2022 08:45:41

%S 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,

%T 55,11,12,54,112,105,36,2,13,65,156,182,91,13,14,77,210,294,196,49,2,

%U 15,90,275,450,378,140,15,16,104,352,660,672,336,64,2,17,119,442,935,1122,714,204,17

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

%C Row sums are A001610(n-1).

%C Triangle A034807 (coefficients of Lucas polynomials) with the first column omitted. - _Philippe Deléham_, Mar 17 2013

%C 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

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

%H G. C. Greubel, <a href="/A157000/b157000.txt">Rows n = 2..100 of triangle, flattened</a>

%F T(n,k) = binomial(n-k,k) + binomial(n-k-1,k-1). - _Bert Seghers_, Mar 02 2014

%e The table starts in row n=2, column k=1 as:

%e 2;

%e 3;

%e 4, 2;

%e 5, 5;

%e 6, 9, 2;

%e 7, 14, 7;

%e 8, 20, 16, 2;

%e 9, 27, 30, 9;

%e 10, 35, 50, 25, 2;

%e 11, 44, 77, 55, 11;

%e 12, 54, 112, 105, 36, 2;

%t 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 *)

%o (PARI) a(n,k)=n*binomial(n-k-1,k-1)/k; \\ _Charles R Greathouse IV_, Jul 10 2011

%o (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

%o (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

%Y Cf. A132460, A113279, A082985, A029635.

%K nonn,easy,tabf

%O 2,1

%A _Roger L. Bagula_, Feb 20 2009

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