A192371 Number of lattice paths from (0,0) to (n,n) using steps (1,1), (0,2), (2,0), (0,3), (3,0).

1, 1, 3, 9, 25, 87, 307, 1113, 4149, 15605, 59201, 225999, 866449, 3333847, 12865335, 49769689, 192945411, 749396493, 2915432049, 11358771965, 44313108627, 173081422997, 676766482917, 2648843996031, 10376891445525, 40685535827325, 159641884780749, 626849029013919, 2463010645910537, 9683604464279235

Offset: 0

Author

Joerg Arndt, Jul 01 2011

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A192368.

Programs

Maple

s := RootOf( (s^3-s-1)*(s-1)+x*s*(4-3*s), s);
ogf := sqrt(4*s-3*s^2)*(s^3-4*s^2+2*s+2)/((2*s^2-s-2)*(3*s^3-6*s^2+4*s-2)*(1-x)):
series(ogf, x=0, 30);  # Mark van Hoeij, Apr 17 2013
# second Maple program:
b:= proc(p) b(p):= `if`(p=[0$2], 1, `if`(min(p[])<0, 0,
      add(b(p-l), l=[[1, 1], [0, 2], [2, 0], [0, 3], [3, 0]])))
    end:
a:= n-> b([n$2]):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 18 2014

Mathematica

b[p_List] := b[p] = If[p == {0, 0}, 1, If[Min[p] < 0, 0, Sum[b[p - l], {l, {{1, 1}, {0, 2}, {2, 0}, {3, 0}, {0, 3}}}]]]; a[n_] := b[{n, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

PARI

/* same as in A092566 but use */
steps=[[1,1], [2,0], [0,2], [3,0], [0,3]];
/* Joerg Arndt, Jun 30 2011 */

Formula

G.f.: sqrt(4*s-3*s^2)*(s^3-4*s^2+2*s+2)/((2*s^2-s-2)*(3*s^3-6*s^2+4*s-2)*(1-x)) where the function s satisfies (s^3-s-1)*(s-1)+x*s*(4-3*s) = 0. - Mark van Hoeij, Apr 17 2013