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A361190 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)}.

%I A361190 #102 Aug 04 2023 10:13:35
%S A361190 1,1,9,153,3579,101630,3288871,116951012,4465824585,180310624841,
%T A361190 7614208325878,333613510494834,15075162152856423,699290488810583617,
%U A361190 33176816563410874752,1605135467691243954419,79003021319962788395355,3947913343912428255683930
%N A361190 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 A361190 Is this the same sequence as A217823?
%H A361190 Alois P. Heinz, <a href="/A361190/b361190.txt">Table of n, a(n) for n = 0..400</a>
%H A361190 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%F A361190 a(n) = A217823(n) for n<=6.
%e A361190 a(0) = 1: (00), 0 steps are made.
%e A361190 a(1) = 1: (00)(11)(20)(10)(00).
%e A361190 a(2) = 9:
%e A361190   (00)(11)(20)(10)(00)(11)(20)(10)(00),
%e A361190   (00)(11)(20)(10)(21)(30)(20)(10)(00),
%e A361190   (00)(11)(20)(10)(21)(11)(20)(10)(00),
%e A361190   (00)(11)(20)(31)(40)(30)(20)(10)(00),
%e A361190   (00)(11)(20)(31)(21)(30)(20)(10)(00),
%e A361190   (00)(11)(20)(31)(21)(11)(20)(10)(00),
%e A361190   (00)(11)(22)(31)(40)(30)(20)(10)(00),
%e A361190   (00)(11)(22)(31)(21)(30)(20)(10)(00),
%e A361190   (00)(11)(22)(31)(21)(11)(20)(10)(00).
%p A361190 b:= proc(n, x, y) option remember; `if`(x+2*y>n, 0,
%p A361190      `if`(n=0, 1, `if`(y>0, b(n-1, x+1, y-1), 0)+
%p A361190      `if`(y<x, b(n-1, x-1, y), 0)+b(n-1, x+1, y+1)))
%p A361190     end:
%p A361190 a:= n-> b(4*n, 0$2):
%p A361190 seq(a(n), n=0..17);
%Y A361190 Cf. A001006, A005789, A026945, A151332 (the same without condition on the diagonal), A217823, A359647.
%K A361190 nonn,walk
%O A361190 0,3
%A A361190 _Alois P. Heinz_, Jul 31 2023