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

0,3

Links

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A279512 A111510 A151277 * A355539 A323686 A285853

Adjacent sequences: A192365 A192366 A192367 * A192369 A192370 A192371

Keyword

nonn

Author

Joerg Arndt, Jul 01 2011

Status

approved