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A364439 a(n) is the number of paths with length 3*n that begin at (0,0), end at (0,0), and do not reach (0,0) at any point in between while 0 <= y <= x at every step, where a path is a sequence of steps in the form (1,1), (1,-1), and (-2,0).

%I A364439 #41 Oct 27 2023 13:05:21
%S A364439 1,1,4,33,367,4844,71597,1147653,19559062,349766457,6502419671,
%T A364439 124822220086,2461515013103,49668479230825,1022258042480874,
%U A364439 21406231023989503,455112508356168561,9807294681518154334,213897254891041613995,4715809234441541498539
%N A364439 a(n) is the number of paths with length 3*n that begin at (0,0), end at (0,0), and do not reach (0,0) at any point in between while 0 <= y <= x at every step, where a path is a sequence of steps in the form (1,1), (1,-1), and (-2,0).
%C A364439 If the constraint is removed that the sequence does not reach (0,0) at any point other than the beginning and end of the sequence, this sequence becomes A005789.
%H A364439 Alois P. Heinz, <a href="/A364439/b364439.txt">Table of n, a(n) for n = 0..707</a>
%H A364439 Marshall Hamon, <a href="/A364439/a364439.c.txt">A364439.c</a>
%F A364439 From _Alois P. Heinz_, Jul 29 2023: (Start)
%F A364439 INVERTi transform of A005789.
%F A364439 a(n) mod 2 = A011655(n+1). (End)
%e A364439 Let A represent the (1,1) step, B represent the (1,-1) step, and C represent the (-2,0) step.
%e A364439 For n = 1, the only valid path is ABC.
%e A364439 For n = 2, the 4 valid paths are AABBCC, AABCBC, ABABCC, ABACBC.
%p A364439 b:= proc(n, l) option remember; `if`(n<1, 1, add((h->
%p A364439       `if`(h[2]>h[1] or h[1]>=n or min(h)<0 or n>1 and h=[0$2],
%p A364439            0, b(n-1, h)))(l-w), w=[[1, 1], [1, -1], [-2, 0]]))
%p A364439     end:
%p A364439 a:= n-> b(3*n-1, [2, 0]):
%p A364439 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 28 2023
%p A364439 # second Maple program:
%p A364439 f:= proc(n) option remember; (3*n)!*mul(i!/(n+i)!, i=0..2) end:	
%p A364439 a:= proc(n) option remember; `if`(n=0, 1,
%p A364439       f(n)-add(f(n-i)*a(i), i=1..n-1))
%p A364439     end:
%p A364439 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 29 2023
%t A364439 f[n_] := f[n] = (3n)!*Product[i!/(n+i)!, {i, 0, 2}];
%t A364439 a[n_] := a[n] = If[n == 0, 1, f[n] - Sum[f[n-i]*a[i], {i, 1, n-1}]];
%t A364439 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Oct 27 2023, after _Alois P. Heinz_ *)
%o A364439 (C) /* See Hamon Link */
%Y A364439 Cf. A005789, A011655.
%K A364439 nonn,walk
%O A364439 0,3
%A A364439 _Marshall Hamon_, Jul 24 2023
%E A364439 More terms from _Alois P. Heinz_, Jul 27 2023