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A370498 Number of paths from (0, 0) to (2n, 2n) in an n X n grid using only steps north, northeast and east (i.e., steps (1, 0), (1, 1), and (0, 1)) and that do not pass through diagonal points with odd coordinates.

%I A370498 #21 Mar 25 2024 09:11:25
%S A370498 1,4,60,1204,27724,691812,18198492,496924692,13951437804,400212569284,
%T A370498 11679079547260,345621250279284,10347645099250060,312857431914558244,
%U A370498 9538937406065229084,292961855077076241108,9054857076142602126636,281439018537947499788676,8791174819979940884130492
%N A370498 Number of paths from (0, 0) to (2n, 2n) in an n X n grid using only steps north, northeast and east (i.e., steps (1, 0), (1, 1), and (0, 1)) and that do not pass through diagonal points with odd coordinates.
%F A370498 a(n) = b(2*n,2*n) where b(n,0) = b(0,m) = 1, b(2n+1,2n+1) = 0 and b(n,m) = b(n-1,m-1) + b(n-1,m) + b(n,m-1) otherwise.
%F A370498 a(n) = ceiling((2/3)*A138462(n)).
%F A370498 a(n) ~ (1 + sqrt(2))^(4*n+1) / (3 * 2^(5/4) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 14 2024
%F A370498 D-finite with recurrence +n*(2*n+1)*a(n) +(-70*n^2+73*n-15)*a(n-1) +(70*n^2-277*n+270)*a(n-2) -(2*n-5)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Mar 25 2024
%p A370498 a:= proc(n) option remember; `if`(n<2, 4^n, ((8*n-6)*(34*n^2-51*n+11)
%p A370498       *a(n-1)-(n-2)*(4*n-1)*(2*n-3)*a(n-2))/(n*(4*n-5)*(2*n+1)))
%p A370498     end:
%p A370498 seq(a(n), n=0..18);  # _Alois P. Heinz_, Feb 20 2024
%t A370498 A[n_, m_] :=  A[n, m] = Which[m*n == 0, 1, m == n && Mod[m, 2] == 1, 0, True, A[n - 1, m] + A[n, m - 1] + A[n - 1, m - 1]]; Table[A[2 n, 2 n], {n, 0, 45}]
%Y A370498 Cf. A001850, A008288, A138462.
%K A370498 nonn
%O A370498 0,2
%A A370498 _José María Grau Ribas_, Feb 20 2024