OFFSET
0,4
COMMENTS
This is to factorials what Fine numbers are to Catalan numbers. There is no known combinatorial interpretation.
The same construction, applied to the central binomials, leads to A126984, apart from signs and the first term. - Peter Luschny, Jul 22 2022
a(n) is the number of permutations of [n] whose number of components is odd minus the number of those permutations with an even number of components. - Peter Luschny, Sep 10 2022
LINKS
David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
FindStat - Combinatorial Statistic Finder, The number of connected components of a permutation.
FORMULA
G.f.: f/(1+2*f) where f is (the g.f. of A000142) - 1.
a(n) = -Sum_{k=1..n} (-1)^k * A059438(n, k) for n >= 1. - Peter Luschny, Sep 10 2022
EXAMPLE
Consider the permutations of [3]: [2,3,1], [3,1,2] and [3,2,1] have 1 component,
[1,3,2] and [2,1,3] have 2 components, and [1,2,3] has three components. Thus 3 - 2 + 1 = 2 = a(3). - Peter Luschny, Sep 10 2022
MAPLE
a:= n-> (f-> coeff(series(f/(1+2*f), x, n+1), x, n))(add(j!*x^j, j=1..n)):
seq(a(n), n=0..23); # Alois P. Heinz, Jul 20 2022
MATHEMATICA
nmax=22; f[x_]:=Sum[i! x^i, {i, nmax}]; CoefficientList[Series[f[x]/(1+2f[x]), {x, 0, nmax}], x] (* Stefano Spezia, Jul 04 2022 *)
PROG
(SageMath)
A = QQ[['t']]
f = A([0] + [factorial(n) for n in range(1, 30)]).O(30)
print(list(f/(1+2*f)))
(SageMath) # Uses function A059438_triangle.
def A355488_list(size):
triangle = A059438_triangle(size)
return [0] + [sum((-1)^k*t for (k, t) in enumerate(row)) for row in triangle]
print(A355488_list(20)) # Peter Luschny, Sep 10 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
F. Chapoton, Jul 04 2022
STATUS
approved