A071801 a(n) = binomial(2n, n) - binomial(n, floor(n/2))^2.

0, 1, 2, 11, 34, 152, 524, 2207, 7970, 32744, 121252, 491988, 1850380, 7455944, 28337976, 113708295, 435443490, 1742630120, 6711230900, 26811568916, 103711749284, 413849297784, 1606464657096, 6405315809516, 24935144010764, 99367486347752

Offset

0,3

Comments

Number of lattice paths in the lattice [0..n] X [0..n] which do not pass through the point (floor(n/2),floor(n/2)). In this case, the "hole" in the lattice is at the point closest to the lattice center.

Links

T. D. Noe, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Lattice path.

Formula

a(n) = A000984(n) - A001405(n)^2.

Also, a(n) = Sum_{m=0..n} binomial(n, m)^2 - binomial(n, floor(n/2))^2.

G.f.: 1/sqrt(1-4*x) + 1/(4*x) - (4*x+1)*EllipticK(4*x)/(2*x*Pi). - Mark van Hoeij, May 01 2013

Maple

A071801:=n->binomial(2*n, n) - binomial(n, floor(n/2))^2: seq(A071801(n), n=0..30); # Wesley Ivan Hurt, Jan 03 2017

Mathematica

Table[Binomial[2n, n] - Binomial[n, Floor[n/2]]^2, {n, 0, 20}]

Prog

(Magma) [Binomial(2*n, n) - Binomial(n, Floor(n/2))^2 : n in [0..40]]; // Wesley Ivan Hurt, Jan 03 2017

Crossrefs

Cf. A000984, A001405, A002894, A071800, A071803.

Sequence in context: A026961 A026971 A323683 * A026981 A027223 A027229

Adjacent sequences: A071798 A071799 A071800 * A071802 A071803 A071804

Keyword

nonn,easy

Author

T. D. Noe, Jun 06 2002

Extensions

More terms from Roger L. Bagula, Aug 28 2006

Edited by N. J. A. Sloane, Oct 08 2006

Status

approved