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A190907 Triangle read by rows: T(n,k) = binomial(n+k, n-k) k! / (floor(k/2)! * floor((k+2)/2)!).

%I A190907 #18 Oct 29 2022 09:09:38
%S A190907 1,1,1,1,3,1,1,6,5,3,1,10,15,21,2,1,15,35,84,18,10,1,21,70,252,90,110,
%T A190907 5,1,28,126,630,330,660,65,35,1,36,210,1386,990,2860,455,525,14,1,45,
%U A190907 330,2772,2574,10010,2275,4200,238,126
%N A190907 Triangle read by rows: T(n,k) = binomial(n+k, n-k) k! / (floor(k/2)! * floor((k+2)/2)!).
%C A190907 The triangle may be regarded as a generalization of the triangle A088617.
%C A190907 A088617(n,k) = binomial(n+k,n-k)*(2*k)$/(k+1);
%C A190907 T(n,k) = binomial(n+k,n-k)*(k)$ /(floor(k/2)+1).
%C A190907 Here n$ denotes the swinging factorial A056040(n). As A088617 is a decomposition of the large Schroeder numbers A006318, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.
%C A190907 T(n,n) = A057977(n) which can be seen as extended Catalan numbers.
%H A190907 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.
%F A190907 T(n,1) = A000217(n). T(n,2) = (n-1)*n*(n+1)*(n+2)/24 (Cf. A000332).
%e A190907 [0]  1
%e A190907 [1]  1,  1
%e A190907 [2]  1,  3,   1
%e A190907 [3]  1,  6,   5,   3
%e A190907 [4]  1, 10,  15,  21,   2
%e A190907 [5]  1, 15,  35,  84,  18,  10
%e A190907 [6]  1, 21,  70, 252,  90, 110,  5
%e A190907 [7]  1, 28, 126, 630, 330, 660, 65, 35
%p A190907 A190907 := (n,k) -> binomial(n+k,n-k)*k!/(floor(k/2)!*floor((k+2)/2)!);
%p A190907 seq(print(seq(A190907(n,k), k=0..n)), n=0..7);
%t A190907 Flatten[Table[Binomial[n+k,n-k] k!/(Floor[k/2]!Floor[(k+2)/2]!),{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, May 05 2012 *)
%Y A190907 Cf. Row sums: A190908; A056040, A085478, A088617, A060693.
%K A190907 nonn,tabl
%O A190907 0,5
%A A190907 _Peter Luschny_, May 24 2011