A096402 - OEIS

%I #27 Nov 07 2018 02:41:53

%S 1,1,1,2,5,14,47,182,786,3774,19974,115236,720038,4846512,34950929,

%T 268836776,2197143724,19013216102,173672030192,1669863067916,

%U 16858620684522,178306120148144,1971584973897417,22748265125187632

%N 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.

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

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

%C 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

%H Arvind Ayyer, Matthieu Josuat-Vergès, Sanjay Ramassamy, <a href="https://arxiv.org/abs/1803.10351">Extensions of partial cyclic orders and consecutive coordinate polytopes</a>, arXiv:1803.10351 [math.CO], 2018.

%H R. Stanley, <a href="http://mathoverflow.net/questions/87801/">A polynomial recurrence involving partial derivatives</a>

%F 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).

%F a(n) = n! * g(0, 1, n+1) where g(a, b, n) = f(a, b, n)/a. - _Michael Somos_, Feb 21 2012

%e f(a,b,1)=a, f(a,b,2)= ab - a^2/2.

%e x + x^2 + x^3 + 2*x^4 + 5*x^5 + 14*x^6 + 47*x^7 + 182*x^8 + 786*x^9 + ...

%Y Cf. A000111.

%K nonn

%O 1,4

%A _Richard Stanley_, Aug 06 2004