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