This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379035 #13 Dec 16 2024 10:47:49
%S A379035 1,3,20,201,2574,38740,654334,12050561,237383562,4935186778,
%T A379035 107233505624,2417430607946,56224895607144,1343171657532430,
%U A379035 32841495349026802,819503313717327041,20820095694805722362,537474654188020125882,14075078600122360679520,373373810133893669112710
%N A379035 Number of intervals in the lattice of Schroeder paths of length 2*n.
%C A379035 a(n) is the number of nested pairs of Schroeder paths of length 2*n.
%H A379035 Alois P. Heinz, <a href="/A379035/b379035.txt">Table of n, a(n) for n = 0..661</a>
%e A379035 The a(2) = 20 pairs of Schroeder paths are:
%e A379035 . __
%e A379035 ____ __/\ /\__ /\/\ / \
%e A379035 ____ ____ ____ ____ ____
%e A379035 .
%e A379035 /\ __ /\
%e A379035 / \ __/\ /\/\ / \ / \
%e A379035 ____ __/\ __/\ __/\ __/\
%e A379035 .
%e A379035 __ /\
%e A379035 /\__ /\/\ / \ / \ /\/\
%e A379035 /\__ /\__ /\__ /\__ /\/\
%e A379035 .
%e A379035 __ /\ __ /\ /\
%e A379035 / \ / \ /__\ /__\ //\\
%e A379035 /\/\ /\/\ / \ / \ / \
%o A379035 (Python)
%o A379035 from collections import defaultdict
%o A379035 def a_gen(n): #generator of terms a(0) to a(n)
%o A379035 D = {(0,0):1}
%o A379035 for k in range(2*n+2):
%o A379035 D2 = defaultdict(int)
%o A379035 for i,j in D:
%o A379035 d = D[(i,j)]
%o A379035 mi,mj = (i-k)%2,(j-k)%2
%o A379035 for a in range(-mi,mi+1):
%o A379035 for b in range(-mj,mj+1):
%o A379035 if 0 <= i+a <= j+b <= 2*n-k+1:
%o A379035 D2[(i+a,j+b)] += d
%o A379035 D = D2
%o A379035 if k%2==1:
%o A379035 yield D[(0,0)]
%Y A379035 Cf. A006318 (number of Schroeder paths), A005700 (number of intervals in the lattice of Dyck paths), A379034 (number of intervals in the lattice of Motzkin paths).
%K A379035 nonn
%O A379035 0,2
%A A379035 _Ludovic Schwob_, Dec 14 2024