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

%I A192368 #19 Oct 14 2019 05:04:14
%S A192368 1,1,6,19,94,396,1870,8541,40284,189274,899260,4281168,20487156,
%T A192368 98299384,473118174,2282322211,11034087438,53443135944,259283934816,
%U A192368 1259795078566,6129223177272,29856164309124,145592506783224,710686739172096,3472285996766556,16979257639328076
%N A192368 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (2,0), (0,2), (1,1).
%H A192368 Alois P. Heinz, <a href="/A192368/b192368.txt">Table of n, a(n) for n = 0..1000</a>
%F A192368 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 A192368 s := RootOf( 16*x*(3*s+1)*s+(s^2-18*s+1)*(s-1), s):
%p A192368 ogf := -16*(3*s+1)*s^(3/2)/(3*s^4+2*s^3-76*s^2+6*s+1):
%p A192368 series(ogf, x=0, 20); # _Mark van Hoeij_, Apr 16 2013
%p A192368 # second Maple program:
%p A192368 b:= proc(x, y) option remember;
%p A192368       `if`(min(x, y)<0, 0, `if`(max(x, y)=0, 1,
%p A192368        b(x-1, y)+b(x-2, y)+b(x, y-2)+b(x-1, y-1)))
%p A192368     end:
%p A192368 a:= n-> b(n$2):
%p A192368 seq(a(n), n=0..35);  # _Alois P. Heinz_, May 16 2017
%t A192368 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 A192368 a[n_] := a[n, n];
%t A192368 a /@ Range[0, 25] (* _Jean-François Alcover_, Oct 14 2019 *)
%o A192368 (PARI) /* same as in A092566 but use */
%o A192368 steps=[[1,0], [2,0], [0,2], [1,1]];
%o A192368 /* _Joerg Arndt_, Jun 30 2011 */
%Y A192368 Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369.
%K A192368 nonn
%O A192368 0,3
%A A192368 _Joerg Arndt_, Jul 01 2011