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

1,2

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

This seems to be the coefficient of t^2 in the quantum periods of some Fano manifolds, namely the generic hypersurface of degree n in the projective space P^n. For the generic quadric in P^2, the quantum period starts 1 + 2*t^2 + 6*t^4 + O(t^5). See page 36 of Coates et al. reference for the number 99000 in case n=5. - F. Chapoton, May 06 2026

Links

Alois P. Heinz, Table of n, a(n) for n = 1..238 (first 50 terms from T. D. Noe)

T. Coates, A. Kasprzyk and T. Prince, Four-Dimensional Fano Toric Complete Intersections, arXiv:1409.5030 [math.AG], 2014.

Eric Weisstein's World of Mathematics, Lattice Path

Formula

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

a(n) = A000680(n) - A001044(n). - Alois P. Heinz, May 06 2026

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}]

Prog

(SageMath)
[factorial(2*n)//(2**n) - factorial(n)**2 for n in range(1, 20)]

Crossrefs

Cf. A000680, A001044.

Row n=2 of A225094.

Sequence in context: A157058 A305693 A357421 * A338514 A123686 A122418

Adjacent sequences: A071795 A071796 A071797 * A071799 A071800 A071801

Keyword

easy,nice,nonn

Author

T. D. Noe, Jun 06 2002

Extensions

More terms from Harvey P. Dale, May 26 2011

Status

approved