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.

1, 4, 60, 1204, 27724, 691812, 18198492, 496924692, 13951437804, 400212569284, 11679079547260, 345621250279284, 10347645099250060, 312857431914558244, 9538937406065229084, 292961855077076241108, 9054857076142602126636, 281439018537947499788676, 8791174819979940884130492

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

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.

a(n) = ceiling((2/3)*A138462(n)).

a(n) ~ (1 + sqrt(2))^(4*n+1) / (3 * 2^(5/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 14 2024

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

Maple

a:= proc(n) option remember; `if`(n<2, 4^n, ((8*n-6)*(34*n^2-51*n+11)
      *a(n-1)-(n-2)*(4*n-1)*(2*n-3)*a(n-2))/(n*(4*n-5)*(2*n+1)))
    end:
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 20 2024

Mathematica

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}]

Crossrefs

Cf. A001850, A008288, A138462.

Sequence in context: A156090 A181418 A208890 * A325154 A013486 A013483

Adjacent sequences: A370495 A370496 A370497 * A370499 A370500 A370501

Keyword

nonn

Author

José María Grau Ribas, Feb 20 2024

Status

approved