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A355488 Expansion of g.f. f/(1+2*f) where f is the g.f. of nonempty permutations.

%I A355488 #34 Apr 25 2023 10:44:20
%S A355488 0,1,0,2,8,48,328,2560,22368,216224,2291456,26430336,329805952,
%T A355488 4429255168,63730438656,978479250944,15972310317056,276292865550336,
%U A355488 5049672714569728,97245533647568896,1968395389124714496,41783552069858877440,928204423021249003520
%N A355488 Expansion of g.f. f/(1+2*f) where f is the g.f. of nonempty permutations.
%C A355488 This is to factorials what Fine numbers are to Catalan numbers. There is no known combinatorial interpretation.
%C A355488 The same construction, applied to the central binomials, leads to A126984, apart from signs and the first term. - _Peter Luschny_, Jul 22 2022
%C A355488 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
%H A355488 David Callan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Callan/callan91.html">Counting Stabilized-Interval-Free Permutations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
%H A355488 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000056">The number of connected components of a permutation.</a>
%F A355488 G.f.: f/(1+2*f) where f is (the g.f. of A000142) - 1.
%F A355488 a(n) = -Sum_{k=1..n} (-1)^k * A059438(n, k) for n >= 1. - _Peter Luschny_, Sep 10 2022
%e A355488 Consider the permutations of [3]: [2,3,1], [3,1,2] and [3,2,1] have 1 component,
%e A355488 [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
%p A355488 a:= n-> (f-> coeff(series(f/(1+2*f), x, n+1), x, n))(add(j!*x^j, j=1..n)):
%p A355488 seq(a(n), n=0..23);  # _Alois P. Heinz_, Jul 20 2022
%t A355488 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 *)
%o A355488 (SageMath)
%o A355488 A = QQ[['t']]
%o A355488 f = A([0] + [factorial(n) for n in range(1,30)]).O(30)
%o A355488 print(list(f/(1+2*f)))
%o A355488 (SageMath) # Uses function A059438_triangle.
%o A355488 def A355488_list(size):
%o A355488     triangle = A059438_triangle(size)
%o A355488     return [0] + [sum((-1)^k*t for (k,t) in enumerate(row)) for row in triangle]
%o A355488 print(A355488_list(20))  # _Peter Luschny_, Sep 10 2022
%Y A355488 Cf. A000108, A003319, A000957, A000142, A052186, A126984, A059438, A122827.
%K A355488 nonn
%O A355488 0,4
%A A355488 _F. Chapoton_, Jul 04 2022