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A329851 Sum of absolute values of n-th differences over all permutations of {0, 1, ..., n}. 0
0, 2, 12, 120, 1320, 17856, 273056, 4772624, 92626944, 1986317024, 46556867456, 1184827221584, 32524270418432, 958020105786536 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) <= ((n+1)! - 2*A131502(n))*A130783(n).

Every term is even because the n-th difference of a permutation and its reversal are the same up to sign.

LINKS

Table of n, a(n) for n=0..13.

EXAMPLE

For n = 2, the second differences of the (2+1)! = 6 permutations of {0,1,2} are:

[0,1,2] ->  [1, 1] ->  0,

[0,2,1] ->  [2,-1] -> -3,

[1,0,2] -> [-1, 2] ->  3,

[1,2,0] ->  [1,-2] -> -3,

[2,0,1] -> [-2, 1] ->  3, and

[2,1,0] -> [-1,-1] ->  0.

The sum of the absolute values of these second differences is 0 + 3 + 3 + 3 + 3 + 0 = 12.

MATHEMATICA

a[n_] := Block[{x, k}, k = CoefficientList[(x - 1)^n, x]; Sum[Abs[k.p], {p, Permutations@ Range[0, n]}]]; Array[a, 10, 0] (* Giovanni Resta, Nov 23 2019 *)

CROSSREFS

Cf. A130783, A131502, A327845.

Sequence in context: A089431 A119701 A286629 * A127112 A003580 A052580

Adjacent sequences:  A329848 A329849 A329850 * A329852 A329853 A329854

KEYWORD

nonn,more

AUTHOR

Peter Kagey, Nov 22 2019

EXTENSIONS

a(10) from Alois P. Heinz, Nov 22 2019

a(11)-a(13) from Giovanni Resta, Nov 23 2019

STATUS

approved

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Last modified June 24 21:50 EDT 2021. Contains 345433 sequences. (Running on oeis4.)