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A112413 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and starting with exactly k UD's, where U=(1,1), D=(1,-1) (0 <= k <= n).

%I A112413 #11 Jan 25 2020 01:44:22
%S A112413 1,0,1,1,0,1,3,1,0,1,9,3,1,0,1,28,9,3,1,0,1,90,28,9,3,1,0,1,297,90,28,
%T A112413 9,3,1,0,1,1001,297,90,28,9,3,1,0,1,3432,1001,297,90,28,9,3,1,0,1,
%U A112413 11934,3432,1001,297,90,28,9,3,1,0,1,41990,11934,3432,1001,297,90,28,9,3,1,0
%N A112413 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and starting with exactly k UD's, where U=(1,1), D=(1,-1) (0 <= k <= n).
%C A112413 All columns, except for initial terms, yield A000245. Row sums yield the Catalan numbers (A000108).
%C A112413 Riordan array ((1-x)*c(x),x), c(x) the g.f. of A000108; equal to A125177*A130595. - _Philippe Deléham_, Dec 08 2009
%F A112413 T(n,k) = c(n-k) - c(n-k-1), where c(n) = binomial(2n, n)/(n+1) is the n-th Catalan number. G.f. = (1-z)*C/(1-tz), where C = (1-sqrt(1-4z))/(2z) is the Catalan function.
%e A112413 T(5,2)=3 because we have UDUDUUDDUD, UDUDUUDUDD and UDUDUUUDDD, where U=(1,1), D=(1,-1).
%e A112413 Triangle begins:
%e A112413    1;
%e A112413    0, 1;
%e A112413    1, 0, 1;
%e A112413    3, 1, 0, 1;
%e A112413    9, 3, 1, 0, 1;
%e A112413   28, 9, 3, 1, 0, 1;
%p A112413 T:=proc(n,k) local c: c:=n->binomial(2*n,n)/(n+1): if k<n then c(n-k)-c(n-k-1) elif k=n then 1 else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y A112413 Cf. A000108, A000245.
%K A112413 nonn,tabl
%O A112413 0,7
%A A112413 _Emeric Deutsch_, Dec 08 2005