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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).
3

%I #30 Oct 27 2023 19:32:49

%S 0,2,54,1944,99000,6966000,655678800,80103945600,12372954249600,

%T 2362712677920000,547235129437920000,151247218046601600000,

%U 49191138900262719360000,18601307697723249058560000,8093164859945489259936000000,4014620173473616480790016000000

%N 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).

%C 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

%H Alois P. Heinz, <a href="/A071798/b071798.txt">Table of n, a(n) for n = 1..238</a> (first 50 terms from T. D. Noe)

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

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

%p a:= proc(n) option remember; `if`(n<3, (n-1)*n,

%p n*((3*n^2-7*n+3)*a(n-1)-(2*n-3)*(n-1)^3*a(n-2))/(n-2))

%p end:

%p seq(a(n), n=1..20); # _Alois P. Heinz_, Apr 26 2013

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

%Y Cf. A000680.

%Y Row n=2 of A225094. - _Alois P. Heinz_, Apr 27 2013

%K easy,nice,nonn

%O 1,2

%A _T. D. Noe_, Jun 06 2002

%E More terms from _Harvey P. Dale_, May 26 2011