A329851 Sum of absolute values of n-th differences over all permutations of {0, 1, ..., n}.

0, 2, 12, 120, 1320, 17856, 273056, 4772624, 92626944, 1986317024, 46556867456, 1184827221584, 32524270418432, 958020105786536

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 *)

Prog

(Python)
from math import comb
from itertools import permutations
def A329851(n):
    c = [-comb(n, i) if i&1 else comb(n, i) for i in range(n+1)]
    return sum(abs(sum(c[i]*p[i] for i in range(n+1))) for p in permutations(range(n+1)) if p[0]<p[-1])<<1 # Chai Wah Wu, Jun 04 2024

Crossrefs

Cf. A130783, A131502, A327845.

Sequence in context: A286629 A361595 A370876 * A127112 A365283 A003580

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