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

%I A192371 #20 May 27 2015 10:34:00
%S A192371 1,1,3,9,25,87,307,1113,4149,15605,59201,225999,866449,3333847,
%T A192371 12865335,49769689,192945411,749396493,2915432049,11358771965,
%U A192371 44313108627,173081422997,676766482917,2648843996031,10376891445525,40685535827325,159641884780749,626849029013919,2463010645910537,9683604464279235
%N A192371 Number of lattice paths from (0,0) to (n,n) using steps (1,1), (0,2), (2,0), (0,3), (3,0).
%H A192371 Alois P. Heinz, <a href="/A192371/b192371.txt">Table of n, a(n) for n = 0..1000</a>
%F A192371 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
%p A192371 s := RootOf( (s^3-s-1)*(s-1)+x*s*(4-3*s), s);
%p A192371 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)):
%p A192371 series(ogf, x=0, 30);  # _Mark van Hoeij_, Apr 17 2013
%p A192371 # second Maple program:
%p A192371 b:= proc(p) b(p):= `if`(p=[0$2], 1, `if`(min(p[])<0, 0,
%p A192371       add(b(p-l), l=[[1, 1], [0, 2], [2, 0], [0, 3], [3, 0]])))
%p A192371     end:
%p A192371 a:= n-> b([n$2]):
%p A192371 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 18 2014
%t A192371 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_ *)
%o A192371 (PARI) /* same as in A092566 but use */
%o A192371 steps=[[1,1], [2,0], [0,2], [3,0], [0,3]];
%o A192371 /* _Joerg Arndt_, Jun 30 2011 */
%Y A192371 Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A192368.
%K A192371 nonn
%O A192371 0,3
%A A192371 _Joerg Arndt_, Jul 01 2011