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 A026780 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k) if 1<=k<=[ n/2 ], else T(n,k)=T(n-1,k-1)+T(n-1,k). 31
 1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 12, 5, 1, 1, 9, 24, 17, 6, 1, 1, 11, 40, 53, 23, 7, 1, 1, 13, 60, 117, 76, 30, 8, 1, 1, 15, 84, 217, 246, 106, 38, 9, 1, 1, 17, 112, 361, 580, 352, 144, 47, 10, 1, 1, 19, 144, 557, 1158, 1178, 496, 191, 57, 11 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n,k) is the number of paths from (0,0) to (k,n-k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j>= 0 and edges (i,i+h)-to-(i+1,i+h+1) for i>=0, h>=0. Also, square array R read by antidiagonals with R(i,j) = T(i+j,i) equal number of paths from (0,0) to (i,j). - Max Alekseyev, Jan 13 2015 LINKS M. A. Alekseyev. On Enumeration of Dyck-Schroeder Paths. Journal of Combinatorial Mathematics and Combinatorial Computing 106 (2018), 59-68. arXiv:1601.06158 FORMULA For n>=2*k, T(n,k) = coefficient of x^k in F(x)*S(x)^(n-2*k). For n<=2*k, T(n,k) = coefficient of x^(n-k) in F(x)*C(x)^(2*k-n). Here C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318, and F(x)=S(x)/(1-x*C(x)*S(x)) is o.g.f. for A026781. - Max Alekseyev, Jan 13 2015 EXAMPLE The array T(n,k) starts with: n=0: 1; n=1: 1, 1; n=2: 1, 3, 1; n=3: 1, 5, 4, 1; n=4: 1, 7, 12, 5, 1; n=5: 1, 9, 24, 17, 6, 1; n=6: 1, 11, 40, 53, 23, 7, 1; ... CROSSREFS Cf. A026787 (row sums),  A026781 (center elements), A249488 (row-reversed version) Sequence in context: A285409 A208510 A131767 * A209421 A320435 A275421 Adjacent sequences:  A026777 A026778 A026779 * A026781 A026782 A026783 KEYWORD nonn,tabl AUTHOR EXTENSIONS Edited by Max Alekseyev, Dec 02 2015 STATUS approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)