OFFSET
1,3
COMMENTS
a(n) is the normalized volume of the convex hull of (classical) parking functions of length n. - Andrés R. Vindas-Meléndez, Jan 13 2023
REFERENCES
Vladimir Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian).
LINKS
Aruzhan Amanbayeva and Danielle Wang, The convex hull of parking functions of length n, Enumer. Comb. Appl. 2 (2022), no. 2, Paper No. S2R10, 1.
Mitsuki Hanada, John Lentfer, and Andrés R. Vindas-Meléndez, Generalized parking function polytopes, arXiv:2212.06885 [math.CO], 2022.
FORMULA
a(2)=1, for n>=3, a(n) = A001499(n) + Sum_{k=1..n-2} (-1)^(k+1)*k!*(C(n,k))^2*(n-k)^k*a(n-k).
a(n) = n!*((n-1)/2^(n-1))*Sum_{i=0..n-2} (2i+1)!!*C(n-2,i)*(2n-1)^(n-i-2). [corrected by John Lentfer, Oct 05 2022]
For n>=2, a(n) = (n!/2^n)*Sum_{i=0..n} (2i-1)*(2i-1)!!*C(n,i)*(2n-1)^(n-i-1).
a(n) = Gamma(3/4)*(sqrt(2)*Pi*e)^(-1/2)*n!*n^(n-1/4)*(1+O(n^((-1/4)+epsilon) with arbitrary small epsilon>0 for sufficiently large n.
MATHEMATICA
Table[n!/2^n * Sum[(2*i-1)*(2*i-1)!!*Binomial[n, i]*(2n-1)^(n-i-1), {i, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Nov 30 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Mar 23 2010
STATUS
approved