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A192364 Number of lattice paths from (0,0) to (n,n) using steps (0,1),(0,2),(1,0),(2,0),(1,1).

%I A192364 #21 Sep 27 2020 15:07:51
%S A192364 1,3,21,157,1239,10047,82951,693603,5854581,49778997,425712429,
%T A192364 3657968097,31555053921,273109567797,2370474720369,20625186298269,
%U A192364 179841473895447,1571088267426447,13747953837604959,120482775658910763,1057293764707074027,9289536349244758791,81709329486947791419
%N A192364 Number of lattice paths from (0,0) to (n,n) using steps (0,1),(0,2),(1,0),(2,0),(1,1).
%H A192364 Seiichi Manyama, <a href="/A192364/b192364.txt">Table of n, a(n) for n = 0..1000</a>
%F A192364 From _Vaclav Kotesovec_, Oct 24 2012: (Start)
%F A192364 G.f.: (3 - 6*x + sqrt(-1 + 4*x*(9*x-11) + 4*sqrt(1-x)*sqrt(5+4*x)*sqrt(9*x-1))) / (sqrt(10+8*x)*sqrt((1-x)*(1-9*x))*(4*x*(9*x-11)-1+4*sqrt(1-x)*sqrt(5+4*x)*sqrt(9*x-1))^(1/4))
%F A192364 D-finite with recurrence: 15*(n-1)*n*a(n) = (n-1)*(133*n-54)*a(n-1) + (31*n^2 - 177*n + 224)*a(n-2) - (113*n^2 - 295*n + 144)*a(n-3) - 18*(n-3)*(2*n-5)*a(n-4)
%F A192364 a(n) ~ 3^(2*n+3/2)/(2*sqrt(14*Pi*n))
%F A192364 (End)
%F A192364 a(n) = A091533(2*n,n) for n >= 0. - _Paul D. Hanna_, Dec 11 2018
%F A192364 a(n) = [x^n*y^n] 1/(1 - x - y - x^2 - x*y - y^2) for n >= 0. - _Paul D. Hanna_, Dec 11 2018
%t A192364 FullSimplify[CoefficientList[Series[(3-6*x+Sqrt[-1+4*x*(9*x-11)+4*Sqrt[1-x]*Sqrt[5+4*x]*Sqrt[9*x-1]])/(Sqrt[10+8*x]*Sqrt[(1-x)*(1-9*x)]*(4*x*(9*x-11)-1+4*Sqrt[1-x]*Sqrt[5+4*x]*Sqrt[9*x-1])^(1/4)), {x, 0, 10}], x]]
%o A192364 (PARI) /* same as in A092566 but use */
%o A192364 steps=[[0,1], [0,2], [1,0], [2,0], [1,1]];
%o A192364 /* _Joerg Arndt_, Jun 30 2011 */
%Y A192364 Cf. A091533.
%K A192364 nonn,walk
%O A192364 0,2
%A A192364 _Eric Werley_, Jun 29 2011
%E A192364 Terms > 425712429 by _Joerg Arndt_, Jun 30 2011