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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). 0
 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, 9, 3, 1, 0, 1, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 3432, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 11934, 3432, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 41990, 11934, 3432, 1001, 297, 90, 28, 9, 3, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS All columns, except for initial terms, yield A000245. Row sums yield the Catalan numbers (A000108). Riordan array ((1-x)*c(x),x), c(x) the g.f. of A000108; equal to A125177*A130595. - Philippe Deléham, Dec 08 2009 LINKS FORMULA 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. EXAMPLE T(5,2)=3 because we have UDUDUUDDUD, UDUDUUDUDD and UDUDUUUDDD, where U=(1,1), D=(1,-1). Triangle begins:    1;    0, 1;    1, 0, 1;    3, 1, 0, 1;    9, 3, 1, 0, 1;   28, 9, 3, 1, 0, 1; MAPLE T:=proc(n, k) local c: c:=n->binomial(2*n, n)/(n+1): if k

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Last modified June 14 12:04 EDT 2021. Contains 345025 sequences. (Running on oeis4.)