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%I #16 Sep 08 2022 08:45:56
%S 1,1,1,1,2,1,1,3,3,3,1,4,6,12,2,1,5,10,30,10,10,1,6,15,60,30,60,5,1,7,
%T 21,105,70,210,35,35,1,8,28,168,140,560,140,280,14,1,9,36,252,252,
%U 1260,420,1260,126,126,1,10,45,360,420,2520,1050,4200,630,1260,42
%N Triangle read by rows: T(n,k) = binomial(n, k) * k! / (floor(k/2)! * floor((k+2)/2)!).
%C The triangle may be regarded a generalization of the triangle A097610:
%C A097610(n,k) = binomial(n,k)*(2*k)$/(k+1);
%C T(n,k) = binomial(n,k)*(k)$/(floor(k/2)+1).
%C Here n$ denotes the swinging factorial A056040(n). As A097610 is a decomposition of the Motzkin numbers A001006, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.
%C T(n,n) = A057977(n) which can be seen as extended Catalan numbers.
%H G. C. Greubel, <a href="/A189913/b189913.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers.</a>
%F From _R. J. Mathar_, Jun 07 2011: (Start)
%F T(n,1) = n.
%F T(n,2) = A000217(n-1).
%F T(n,3) = A027480(n-2).
%F T(n,4) = A034827(n). (End)
%e [0] 1
%e [1] 1, 1
%e [2] 1, 2, 1
%e [3] 1, 3, 3, 3
%e [4] 1, 4, 6, 12, 2
%e [5] 1, 5, 10, 30, 10, 10
%e [6] 1, 6, 15, 60, 30, 60, 5
%e [7] 1, 7, 21, 105, 70, 210, 35, 35
%p A189913 := (n,k) -> binomial(n,k)*(k!/iquo(k,2)!^2)/(iquo(k,2)+1):
%p seq(print(seq(A189913(n,k),k=0..n)),n=0..7);
%t T[n_, k_] := Binomial[n, k]*k!/((Floor[k/2])!*(Floor[(k + 2)/2])!); Table[T[n, k], {n, 0, 10}, {k, 0, n}]// Flatten (* _G. C. Greubel_, Jan 13 2018 *)
%o (PARI) {T(n,k) = binomial(n,k)*k!/((floor(k/2))!*(floor((k+2)/2))!) };
%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jan 13 2018
%o (Magma) /* As triangle */ [[Binomial(n,k)*Factorial(k)/(Factorial(Floor(k/2))*Factorial(Floor((k + 2)/2))): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Jan 13 2018
%Y Row sums are A189912.
%Y Cf. A097610, A057977, A001006.
%K nonn,tabl,easy
%O 0,5
%A _Peter Luschny_, May 24 2011
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