A192368 - OEIS

%I #19 Oct 14 2019 05:04:14

%S 1,1,6,19,94,396,1870,8541,40284,189274,899260,4281168,20487156,

%T 98299384,473118174,2282322211,11034087438,53443135944,259283934816,

%U 1259795078566,6129223177272,29856164309124,145592506783224,710686739172096,3472285996766556,16979257639328076

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

%H Alois P. Heinz, <a href="/A192368/b192368.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

%p s := RootOf( 16*x*(3*s+1)*s+(s^2-18*s+1)*(s-1), s):

%p ogf := -16*(3*s+1)*s^(3/2)/(3*s^4+2*s^3-76*s^2+6*s+1):

%p series(ogf, x=0, 20); # _Mark van Hoeij_, Apr 16 2013

%p # second Maple program:

%p b:= proc(x, y) option remember;

%p       `if`(min(x, y)<0, 0, `if`(max(x, y)=0, 1,

%p        b(x-1, y)+b(x-2, y)+b(x, y-2)+b(x-1, y-1)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..35);  # _Alois P. Heinz_, May 16 2017

%t 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;

%t a[n_] := a[n, n];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Oct 14 2019 *)

%o (PARI) /* same as in A092566 but use */

%o steps=[[1,0], [2,0], [0,2], [1,1]];

%o /* _Joerg Arndt_, Jun 30 2011 */

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

%K nonn

%O 0,3

%A _Joerg Arndt_, Jul 01 2011