A379035 Number of intervals in the lattice of Schroeder paths of length 2*n.
1, 3, 20, 201, 2574, 38740, 654334, 12050561, 237383562, 4935186778, 107233505624, 2417430607946, 56224895607144, 1343171657532430, 32841495349026802, 819503313717327041, 20820095694805722362, 537474654188020125882, 14075078600122360679520, 373373810133893669112710
Offset: 0
Keywords
Examples
The a(2) = 20 pairs of Schroeder paths are:
. __
____ __/\ /\__ /\/\ / \
____ ____ ____ ____ ____
.
/\ __ /\
/ \ __/\ /\/\ / \ / \
____ __/\ __/\ __/\ __/\
.
__ /\
/\__ /\/\ / \ / \ /\/\
/\__ /\__ /\__ /\__ /\/\
.
__ /\ __ /\ /\
/ \ / \ /__\ /__\ //\\
/\/\ /\/\ / \ / \ / \
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..661
Crossrefs
Programs
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Python
from collections import defaultdict def a_gen(n): #generator of terms a(0) to a(n) D = {(0,0):1} for k in range(2*n+2): D2 = defaultdict(int) for i,j in D: d = D[(i,j)] mi,mj = (i-k)%2,(j-k)%2 for a in range(-mi,mi+1): for b in range(-mj,mj+1): if 0 <= i+a <= j+b <= 2*n-k+1: D2[(i+a,j+b)] += d D = D2 if k%2==1: yield D[(0,0)]
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