

A026780


Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n1, T(n,k)=T(n1,k1)+T(n2,k1)+T(n1,k) if 1<=k<=[ n/2 ], else T(n,k)=T(n1,k1)+T(n1,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
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OFFSET

0,5


COMMENTS

T(n,k) is the number of paths from (0,0) to (k,nk) 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

Table of n, a(n) for n=0..64.
M. A. Alekseyev. On Enumeration of DyckSchroeder Paths. Journal of Combinatorial Mathematics and Combinatorial Computing 106 (2018), 5968. arXiv:1601.06158


FORMULA

For n>=2*k, T(n,k) = coefficient of x^k in F(x)*S(x)^(n2*k). For n<=2*k, T(n,k) = coefficient of x^(nk) in F(x)*C(x)^(2*kn). Here C(x)=(1sqrt(14x))/(2*x) is o.g.f. for A000108, S(x)=(1xsqrt(16*x+x^2))/(2*x) is o.g.f. for A006318, and F(x)=S(x)/(1x*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 (rowreversed version)
Sequence in context: A285409 A208510 A131767 * A209421 A320435 A275421
Adjacent sequences: A026777 A026778 A026779 * A026781 A026782 A026783


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling


EXTENSIONS

Edited by Max Alekseyev, Dec 02 2015


STATUS

approved



