OFFSET
0,5
COMMENTS
Similarly, 1/(1 - x + x^2/2! - ... - x^(2m-1)/(2m-1)!) is the e.g.f. for permutations in which every increasing run has length 0 or 1 (mod 2m).
Also the 3-degree-chromatic polynomial of the complete graph K_n when at lambda = -1. - Medet Jumadildayev, Jan 11 2026
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..485
David Galvin, John Engbers, and Clifford Smyth, Reciprocals of thinned exponential series, arXiv:2303.14057 [math.CO], 2023.
Ira M. Gessel, Reciprocals of exponential polynomials and permutation enumeration, arXiv:1807.09290 [math.CO], 2018.
Medet Jumadildayev, Increasing Trees and the Degree-Chromatic Polynomial, Kazakh Math. Journal, 4 (2025), 57-66.
FORMULA
E.g.f.: 1/(1 - x + x^2/2! - x^3/3!).
a(0) = a(1) = a(2) = 1; a(n) = n * a(n-1) - n * (n-1) * a(n-2) / 2 + n * (n-1) * (n-2) * a(n-3) / 6 for n > 2. - Ilya Gutkovskiy, Jan 22 2024
a(n) = Sum_{k=1..n} (-1)^k*k!*B(n,k,[1,1,1,0,0,0,...]), where B(n,k,[x1,x2,...]) is the partial Bell polynomial. - Medet Jumadildayev, Jan 11 2026
EXAMPLE
For n=4 the a(4)=2 permutations are 4321 and 1234.
MAPLE
gser:=series(1/(1-x+x^2/2!-x^3/3!), x, 21): seq(n!*coeff(gser, x, n), n=0..20);
MATHEMATICA
With[{nmax = 25}, CoefficientList[Series[1/(1 -x +x^2/2! -x^3/3!), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 30 2018 *)
PROG
(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1 -x +x^2/2 -x^3/6))) \\ G. C. Greubel, Nov 30 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/(1-x+x^2/2-x^3/6) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Nov 30 2018
(SageMath)
f= 1/(1 -x +x^2/2 -x^3/6)
g=f.taylor(x, 0, 13)
L=g.coefficients()
coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
coeffs # G. C. Greubel, Nov 30 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ira M. Gessel, Jul 21 2018
STATUS
approved