A192368 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (2,0), (0,2), (1,1).

1, 1, 6, 19, 94, 396, 1870, 8541, 40284, 189274, 899260, 4281168, 20487156, 98299384, 473118174, 2282322211, 11034087438, 53443135944, 259283934816, 1259795078566, 6129223177272, 29856164309124, 145592506783224, 710686739172096, 3472285996766556, 16979257639328076

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.

Programs

Maple

s := RootOf( 16*x*(3*s+1)*s+(s^2-18*s+1)*(s-1), s):
ogf := -16*(3*s+1)*s^(3/2)/(3*s^4+2*s^3-76*s^2+6*s+1):
series(ogf, x=0, 20); # Mark van Hoeij, Apr 16 2013
# second Maple program:
b:= proc(x, y) option remember;
      `if`(min(x, y)<0, 0, `if`(max(x, y)=0, 1,
       b(x-1, y)+b(x-2, y)+b(x, y-2)+b(x-1, y-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, May 16 2017

Mathematica

a[0, 0] = 1; a[n_, k_] /; n >= 0 && k >= 0 := a[n, k] = a[n, k - 1] + a[n, k - 2] + a[n - 1, k - 1] + a[n - 2, k]; a[, ] = 0;
a[n_] := a[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Oct 14 2019 *)

PARI

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

Formula

G.f. -16*(3*s+1)*s^(3/2)/(3*s^4+2*s^3-76*s^2+6*s+1) where s satisfies 16*x*(3*s+1)*s+(s^2-18*s+1)*(s-1) = 0. - Mark van Hoeij, Apr 16 2013