A190909 - OEIS

%I #12 Jul 23 2017 03:53:33

%S 1,1,1,1,3,2,1,6,10,6,1,10,30,42,6,1,15,70,168,54,30,1,21,140,504,270,

%T 330,20,1,28,252,1260,990,1980,260,140,1,36,420,2772,2970,8580,1820,

%U 2100,70,1,45,660,5544,7722,30030,9100,16800,1190,630

%N Triangle read by rows: T(n,k) = binomial(n+k,n-k) * k! / floor(k/2)!^2.

%C The triangle may be regarded as a generalization of the triangle A063007.

%C A063007(n,k) = binomial(n+k, n-k)*(2*k)$;

%C T(n,k) = binomial(n+k, n-k)*(k)$.

%C Here n$ denotes the swinging factorial A056040(n). As A063007 is a decomposition of the central Delannoy numbers A001850, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.

%C T(n,n) = A056040(n) which can be seen as extended central binomial numbers.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>

%H R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects counted by the central Delannoy numbers</a>, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

%F T(n,1) = A000217(n). T(n,2) = 2*binomial(n+2,4) (Cf. A034827).

%e [0]  1

%e [1]  1,  1

%e [2]  1,  3,   2

%e [3]  1,  6,  10,    6

%e [4]  1, 10,  30,   42,   6

%e [5]  1, 15,  70,  168,  54,   30

%e [6]  1, 21, 140,  504, 270,  330,  20

%e [7]  1, 28, 252, 1260, 990, 1980, 260, 140

%p A190909 := (n,k) -> binomial(n+k,n-k)*k!/iquo(k,2)!^2:

%p seq(print(seq(A190909(n,k),k=0..n)),n=0..7);

%t Flatten[Table[Binomial[n+k,n-k] k!/(Floor[k/2]!)^2,{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Mar 25 2012 *)

%Y Cf. Row sums: A190910; A056040, A063007, A085478, A088617.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, May 24 2011