A071798 Number of paths on the surface of the n-dimensional lattice [0..2]^n; i.e., the lattice paths that do not pass through the point (1,1,...,1).

0, 2, 54, 1944, 99000, 6966000, 655678800, 80103945600, 12372954249600, 2362712677920000, 547235129437920000, 151247218046601600000, 49191138900262719360000, 18601307697723249058560000, 8093164859945489259936000000, 4014620173473616480790016000000

Offset: 1

Author

T. D. Noe, Jun 06 2002

Keywords

• easy
• nice
• nonn

Comments

a(2) + 1 = 3 is prime. a(3) - 1 = 53 is prime. a(5) - 1 = 98999 is prime. a(7) + 1 = 655678801 is prime. a(8) - 1 = 80103945599 is prime, and part of a twin prime, as a(8) + 1 = 80103945601 is prime. a(13) - 1 = 49191138900262719359999 is prime. - Jonathan Vos Post, Sep 01 2009

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..238 (first 50 terms from T. D. Noe)
• Eric Weisstein's World of Mathematics, Lattice Path

Crossrefs

Cf. A000680.

Row n=2 of A225094. - Alois P. Heinz, Apr 27 2013

Programs

Maple

a:= proc(n) option remember; `if`(n<3, (n-1)*n,
       n*((3*n^2-7*n+3)*a(n-1)-(2*n-3)*(n-1)^3*a(n-2))/(n-2))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Apr 26 2013

Mathematica

Table[(2n)!/2^n-(n!)^2, {n, 10}]

Formula

a(n) = (2n)!/2^n - (n!)^2.

Extensions

More terms from Harvey P. Dale, May 26 2011