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