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A318047 a(n) = sum of values taken by all parking functions of length n. 2
1, 8, 81, 1028, 15780, 284652, 5903464, 138407544, 3619892160, 104485268960, 3299177464704, 113120695539612, 4185473097734656, 166217602768452900, 7051983744002135040, 318324623296131263408, 15232941497754507165696, 770291040239888149405944, 41042353622873800536064000, 2298206207793743728251532020 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified June 25 08:27 EDT 2021. Contains 345453 sequences. (Running on oeis4.)