A071801 - OEIS

%I #28 Sep 08 2022 08:45:06

%S 0,1,2,11,34,152,524,2207,7970,32744,121252,491988,1850380,7455944,

%T 28337976,113708295,435443490,1742630120,6711230900,26811568916,

%U 103711749284,413849297784,1606464657096,6405315809516,24935144010764,99367486347752

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

%C 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.

%H T. D. Noe, <a href="/A071801/b071801.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatticePath.html">Lattice path.</a>

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

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

%F 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

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

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

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

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

%K nonn,easy

%O 0,3

%A _T. D. Noe_, Jun 06 2002

%E More terms from _Roger L. Bagula_, Aug 28 2006

%E Edited by _N. J. A. Sloane_, Oct 08 2006