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, 4, 33, 367, 4844, 71597, 1147653, 19559062, 349766457, 6502419671, 124822220086, 2461515013103, 49668479230825, 1022258042480874, 21406231023989503, 455112508356168561, 9807294681518154334, 213897254891041613995, 4715809234441541498539
Offset: 0
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..707
- Marshall Hamon, A364439.c
Programs
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C
/* See Hamon Link */
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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, f(n)-add(f(n-i)*a(i), i=1..n-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 *)
Formula
From Alois P. Heinz, Jul 29 2023: (Start)
INVERTi transform of A005789.
a(n) mod 2 = A011655(n+1). (End)
Extensions
More terms from Alois P. Heinz, Jul 27 2023
Comments