

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 <= n2.


0



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

1,4


COMMENTS

The problem of computing the polytope volume was raised by A. N. Kirillov.
Stanley refers to Exercise4.56(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 <= n3, 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 threedimensional version of the boustrophedon, which in its classical twodimensional form is used to compute the Euler zigzag numbers A000111, see my paper with Ayyer and JosuatVergès linked below.  Sanjay Ramassamy, Nov 03 2018


LINKS

Table of n, a(n) for n=1..24.
Arvind Ayyer, Matthieu JosuatVergès, 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(ba, 1a, n1).
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: A115276 A327702 A317784 * A007268 A326898 A109156
Adjacent sequences: A096399 A096400 A096401 * A096403 A096404 A096405


KEYWORD

nonn


AUTHOR

Richard Stanley, Aug 06 2004


STATUS

approved



