A096402 n! times the volume of the polytope x_i >= 0 for 1 <= i <= n and x_i + x_{i+1} + x_{i+2} <= 1 for 1 <= i <= n-2.

1, 1, 1, 2, 5, 14, 47, 182, 786, 3774, 19974, 115236, 720038, 4846512, 34950929, 268836776, 2197143724, 19013216102, 173672030192, 1669863067916, 16858620684522, 178306120148144, 1971584973897417, 22748265125187632

Offset

1,4

Comments

The problem of computing the polytope volume was raised by A. N. Kirillov.

Stanley refers to Exercise-4.56(d) of Enumerative Combinatorics, vol. 1, 2nd ed. in mathoverflow question 87801. - Michael Somos, Feb 07 2012

Number of ways of placing the numbers {0,1,...,n} on a circle so that for any 0 <= i <= n-3, starting from i and turning in the positive direction, one encounters first i+1, then i+2, then i+3 before returning to i. These numbers can be computed using a three-dimensional version of the boustrophedon, which in its classical two-dimensional form is used to compute the Euler zigzag numbers A000111, see my paper with Ayyer and Josuat-Vergès linked below. - Sanjay Ramassamy, Nov 03 2018

Links

Table of n, a(n) for n=1..24.

Arvind Ayyer, Matthieu Josuat-Vergès, Sanjay Ramassamy, Extensions of partial cyclic orders and consecutive coordinate polytopes, arXiv:1803.10351 [math.CO], 2018.

R. Stanley, A polynomial recurrence involving partial derivatives

Formula

f(1, 1, n)*n!, where f(a, b, 0)=1, f(0, b, n) = 0 for n>0 and the derivative of f(a, b, n) with respect to a is f(b-a, 1-a, n-1).

a(n) = n! * g(0, 1, n+1) where g(a, b, n) = f(a, b, n)/a. - Michael Somos, Feb 21 2012

Example

f(a,b,1)=a, f(a,b,2)= ab - a^2/2.
x + x^2 + x^3 + 2*x^4 + 5*x^5 + 14*x^6 + 47*x^7 + 182*x^8 + 786*x^9 + ...

Crossrefs

Cf. A000111.

Sequence in context: A343664 A327702 A317784 * A007268 A326898 A109156

Adjacent sequences: A096399 A096400 A096401 * A096403 A096404 A096405

Keyword

nonn

Author

Richard Stanley, Aug 06 2004

Status

approved