OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..100
Y. Yao and D. Zeilberger, An Experimental Mathematics Approach to the Area Statistics of Parking Functions, arXiv 1806.02680, 2018
FORMULA
a(n) is the first derivative of P(n,1,x) evaluated at x = 1 where P(n,m,x) satisfies P(n,m,x) = x^n*Sum_{k=0..n} binomial(n,k)*P(n-k, m+k-1, x) with P(0,m,x) = 1 and P(n,0,x) = 0 for n > 0.
a(n) = Sum_{k=1..n} k*A298593(n, k). - Andrew Howroyd, Aug 17 2018
EXAMPLE
Case n = 2: There are 3 parking functions of length 2: [1, 1], [1, 2], [2, 1]. Summing up all values gives 2 + 3 + 3 = 8, so a(2) = 8.
Case n = 3: There are 16 parking functions of length 3: [1, 1, 1], [1, 1, 2], [1, 1, 3], [1, 2, 1], [1, 2, 2], [1, 2, 3], [1, 3, 1], [1, 3, 2], [2, 1, 1], [2, 1, 2], [2, 1, 3], [2, 2, 1], [2, 3, 1], [3, 1, 1], [3, 1, 2], [3, 2, 1]. Summing up all values gives a total of 81, so a(3) = 81.
MAPLE
#Pnax(n, a, x): the sum of x^(sum of all entries in the parking function) over the set of a-parking functions of length n by recurrence relation.
Pnax:=proc(n, a, x) local k:
option remember:
if n=0 then
return 1:
fi:
if n>0 and a=0 then
return 0:
fi:
return expand(x^n*add(binomial(n, k)*Pnax(n-k, a+k-1, x), k=0..n)):
end:
seq(subs(x = 1, diff(Pnax(n, 1, x), x)), n = 1 .. 20)
MATHEMATICA
T[n_, k_] := n Sum[Binomial[n-1, j-1] j^(j-2) (n-j+1)^(n-j-1), {j, k, n}];
a[n_] := Sum[k T[n, k], {k, 1, n}];
Array[a, 20] (* Jean-François Alcover, Aug 29 2018, after Andrew Howroyd *)
PROG
(PARI) \\ here T(n, k) is A298593.
T(n, k)={n*sum(j=k, n, binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j))}
a(n)={sum(k=1, n, k*T(n, k))} \\ Andrew Howroyd, Aug 17 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Yukun Yao, Aug 13 2018
EXTENSIONS
Edited by Andrew Howroyd and N. J. A. Sloane, Aug 19 2018
STATUS
approved