A071803 -id:A071803 - cp's OEIS frontend

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

T. D. Noe, Jun 06 2002

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.

T. D. Noe, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lattice path.

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

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

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

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

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

More terms from Roger L. Bagula, Aug 28 2006

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

A071800 Number of lattice paths in the lattice [0..2n] X [0..2n] X [0..2n] which do not pass through the point (n,n,n). Number of paths through a lattice containing a "hole".

Original entry on oeis.org

54, 26550, 14330736, 8264889270, 4978317147804, 3090501687886704, 1961313438507566400, 1265749551338006549430, 827693426702217868006500, 547017649101008848332870300, 364691794467483796757326646400, 244920478151771001164678945670000

Offset: 1

T. D. Noe, Jun 06 2002

Eric Weisstein's World of Mathematics, Lattice Path.

Cf. A006480, A071801, A071803.

Table[Factorial[6n]/Factorial[2n]^3-(Factorial[3n]/Factorial[n]^3)^2, {n, 1, 10}]

a(n) = s(3,2n)-s(3,n)^2.

cp's OEIS Frontend

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

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