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

%I A157000 #25 Sep 08 2022 08:45:41
%S A157000 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 A157000 55,11,12,54,112,105,36,2,13,65,156,182,91,13,14,77,210,294,196,49,2,
%U A157000 15,90,275,450,378,140,15,16,104,352,660,672,336,64,2,17,119,442,935,1122,714,204,17
%N A157000 Triangle T(n,k) = (n/k)*binomial(n-k-1, k-1) read by rows.
%C A157000 Row sums are A001610(n-1).
%C A157000 Triangle A034807 (coefficients of Lucas polynomials) with the first column omitted. - _Philippe Deléham_, Mar 17 2013
%C A157000 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 A157000 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 199
%H A157000 G. C. Greubel, <a href="/A157000/b157000.txt">Rows n = 2..100 of triangle, flattened</a>
%F A157000 T(n,k) = binomial(n-k,k) + binomial(n-k-1,k-1). - _Bert Seghers_, Mar 02 2014
%e A157000 The table starts in row n=2, column k=1 as:
%e A157000    2;
%e A157000    3;
%e A157000    4,  2;
%e A157000    5,  5;
%e A157000    6,  9,   2;
%e A157000    7, 14,   7;
%e A157000    8, 20,  16,   2;
%e A157000    9, 27,  30,   9;
%e A157000   10, 35,  50,  25,  2;
%e A157000   11, 44,  77,  55, 11;
%e A157000   12, 54, 112, 105, 36, 2;
%t A157000 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 A157000 (PARI) a(n,k)=n*binomial(n-k-1,k-1)/k; \\ _Charles R Greathouse IV_, Jul 10 2011
%o A157000 (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 A157000 (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 A157000 Cf. A132460, A113279, A082985, A029635.
%K A157000 nonn,easy,tabf
%O A157000 2,1
%A A157000 _Roger L. Bagula_, Feb 20 2009
%E A157000 Offset 2, keyword:tabf, more terms by the Assoc. Eds. of the OEIS, Nov 01 2010