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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).
1
1, 1, 4, 33, 367, 4844, 71597, 1147653, 19559062, 349766457, 6502419671, 124822220086, 2461515013103, 49668479230825, 1022258042480874, 21406231023989503, 455112508356168561, 9807294681518154334, 213897254891041613995, 4715809234441541498539
OFFSET
0,3
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.
Marshall Hamon, A364439.c
FORMULA
From Alois P. Heinz, Jul 29 2023: (Start)
INVERTi transform of A005789.
a(n) mod 2 = A011655(n+1). (End)
EXAMPLE
Let A represent the (1,1) step, B represent the (1,-1) step, and C represent the (-2,0) step.
For n = 1, the only valid path is ABC.
For n = 2, the 4 valid paths are AABBCC, AABCBC, ABABCC, ABACBC.
MAPLE
b:= proc(n, l) option remember; `if`(n<1, 1, add((h->
`if`(h[2]>h[1] or h[1]>=n or min(h)<0 or n>1 and h=[0\$2],
0, b(n-1, h)))(l-w), w=[[1, 1], [1, -1], [-2, 0]]))
end:
a:= n-> b(3*n-1, [2, 0]):
seq(a(n), n=0..25); # Alois P. Heinz, Jul 28 2023
# second Maple program:
f:= proc(n) option remember; (3*n)!*mul(i!/(n+i)!, i=0..2) end:
a:= proc(n) option remember; `if`(n=0, 1,
end:
seq(a(n), n=0..25); # Alois P. Heinz, Jul 29 2023
MATHEMATICA
f[n_] := f[n] = (3n)!*Product[i!/(n+i)!, {i, 0, 2}];
a[n_] := a[n] = If[n == 0, 1, f[n] - Sum[f[n-i]*a[i], {i, 1, n-1}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 27 2023, after Alois P. Heinz *)
PROG
(C) /* See Hamon Link */
CROSSREFS
Sequence in context: A362604 A075132 A303919 * A208961 A113170 A187738
KEYWORD
nonn,walk
AUTHOR
Marshall Hamon, Jul 24 2023
EXTENSIONS
More terms from Alois P. Heinz, Jul 27 2023
STATUS
approved

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Last modified September 20 08:53 EDT 2024. Contains 376067 sequences. (Running on oeis4.)