A361190 - OEIS

%I #102 Aug 04 2023 10:13:35

%S 1,1,9,153,3579,101630,3288871,116951012,4465824585,180310624841,

%T 7614208325878,333613510494834,15075162152856423,699290488810583617,

%U 33176816563410874752,1605135467691243954419,79003021319962788395355,3947913343912428255683930

%N Number of 4n-step lattice paths starting and ending at (0,0) that do not go above the diagonal x=y or below the x-axis using steps in {(1,1), (1,-1), (-1,0)}.

%C Is this the same sequence as A217823?

%H Alois P. Heinz, <a href="/A361190/b361190.txt">Table of n, a(n) for n = 0..400</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>

%F a(n) = A217823(n) for n<=6.

%e a(0) = 1: (00), 0 steps are made.

%e a(1) = 1: (00)(11)(20)(10)(00).

%e a(2) = 9:

%e   (00)(11)(20)(10)(00)(11)(20)(10)(00),

%e   (00)(11)(20)(10)(21)(30)(20)(10)(00),

%e   (00)(11)(20)(10)(21)(11)(20)(10)(00),

%e   (00)(11)(20)(31)(40)(30)(20)(10)(00),

%e   (00)(11)(20)(31)(21)(30)(20)(10)(00),

%e   (00)(11)(20)(31)(21)(11)(20)(10)(00),

%e   (00)(11)(22)(31)(40)(30)(20)(10)(00),

%e   (00)(11)(22)(31)(21)(30)(20)(10)(00),

%e   (00)(11)(22)(31)(21)(11)(20)(10)(00).

%p b:= proc(n, x, y) option remember; `if`(x+2*y>n, 0,

%p      `if`(n=0, 1, `if`(y>0, b(n-1, x+1, y-1), 0)+

%p      `if`(y<x, b(n-1, x-1, y), 0)+b(n-1, x+1, y+1)))

%p     end:

%p a:= n-> b(4*n, 0$2):

%p seq(a(n), n=0..17);

%Y Cf. A001006, A005789, A026945, A151332 (the same without condition on the diagonal), A217823, A359647.

%K nonn,walk

%O 0,3

%A _Alois P. Heinz_, Jul 31 2023