OFFSET
0,3
COMMENTS
The barycenter or signcenter of a permutation is the sum of the signs of the difference between initial and final positions of the objects.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 151 terms from Maxwell Jiang)
FORMULA
a(n) = Sum_{k=0..floor(n/2)} binomial(n, n-2*k)*A320337(k). - Maxwell Jiang, Dec 19 2018 (added by editors)
a(n) ~ sqrt(3) * (1 + exp(-2)*(-1)^n) * n^n / exp(n). - Vaclav Kotesovec, Oct 29 2020
EXAMPLE
(4,1,3,5,2) has difference (3,-1,0,1,-3) and signs (1,-1,0,1,-1) with total 0.
MAPLE
b:= proc(s, t) option remember; (n-> `if`(abs(t)>n, 0, `if`(n=0, 1,
add(b(s minus {j}, t+signum(n-j)), j=s))))(nops(s))
end:
a:= n-> b({$1..n}, 0):
seq(a(n), n=0..14); # Alois P. Heinz, Jul 31 2018
MATHEMATICA
E1[n_ /; n >= 0, 0] = 1;
E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
b[n_] := Sum[(-1)^(n-k) E1[n+k, n] Binomial[2n, n-k], {k, 0, n}];
a[n_] := Sum[Binomial[n, n-2k] b[k], {k, 0, n/2}];
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
Olivier Gérard, Jun 26 2001
EXTENSIONS
One more term from Vladeta Jovovic, Jun 28 2001
a(11)-a(14) from Hugo Pfoertner, Sep 23 2004
a(15)-a(18) from R. H. Hardin, Jul 18 2010
a(19)-a(22) from Kyle G Hess, Jul 30 2018
a(0)=1 prepended by Alois P. Heinz, Jul 30 2018
Terms a(23) and beyond from Maxwell Jiang, Dec 19 2018
STATUS
approved