A318047 a(n) = sum of values taken by all parking functions of length n.

1, 8, 81, 1028, 15780, 284652, 5903464, 138407544, 3619892160, 104485268960, 3299177464704, 113120695539612, 4185473097734656, 166217602768452900, 7051983744002135040, 318324623296131263408, 15232941497754507165696, 770291040239888149405944, 41042353622873800536064000, 2298206207793743728251532020

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

Cf. A000272, A298593.

Sequence in context: A007778 A065440 A338694 * A338685 A092366 A022519

Adjacent sequences: A318044 A318045 A318046 * A318048 A318049 A318050

Keyword

nonn

Author

Yukun Yao, Aug 13 2018

Extensions

Edited by Andrew Howroyd and N. J. A. Sloane, Aug 19 2018

Status

approved