OFFSET
0,4
COMMENTS
Also, number of stablized-interval-free permutations on [n] (see Callan link).
Coefficients in the series reversal of the asymptotic expansion of exp(-x)*Ei(x) for x -> inf, where Ei(x) is the exponential integral. - Vladimir Reshetnikov, Apr 24 2016
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
F. Ardila, F. Rincón and L. Williams, Positroids and non-crossing partitions, arXiv preprint arXiv:1308.2698 [math.CO], 2013.
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
David Callan, Counting stabilized-interval-free permutations, arXiv:math/0310157 [math.CO], 2003.
David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
Colin Defant and Nathan Williams, Coxeter Pop-Tsack Torsing, arXiv:2106.05471 [math.CO], 2021.
Jesse Elliott, Asymptotic expansions of the prime counting function, arXiv:1809.06633 [math.NT], 2018.
Hyungju Park, An Asymptotic Formula for the Number of Stabilized-Interval-Free Permutations, J. Int. Seq. (2023) Vol. 26, Art. 23.9.3.
FORMULA
a(0)=a(1)=1, a(n) = (n-1)*a(n-1) + Sum_{j=2..n-2}(j-1)*a(j)*a(n-j), n >= 2. - David Callan
G.f.: A(x) = x/series_reversion(x*G(x)); G(x) = A(x*G(x)); A(x) = G(x/A(x)); where G(x) is the g.f. of the factorials (A000142). - Paul D. Hanna, Jul 09 2006
G.f.: A(x) = 1 + x/(1 - x*A'(x)/A(x)) = 1 + x/(1-x - x^2*d/dx[(A(x) - 1)/x)]).
G.f.: A(x) = 1 + x*F(x) where F(x) satisfies F(x) = 1 + x*F(x) + x^2*F(x)*F'(x) and F'(x) = d/dx F(x). - Paul D. Hanna, Sep 02 2008
a(n) ~ exp(-1) * n! * (1 - 1/n - 5/(2*n^2) - 32/(3*n^3) - 1643/(24*n^4) - 23017/(40*n^5) - 4215719/(720*n^6)). - Vaclav Kotesovec, Feb 22 2014
A003319(n+1) = coefficient of x^n in A(x)^n. - Michael Somos, Feb 23 2014
EXAMPLE
At n=7, the 7th term of A(x)^7 is 7! x^6, as demonstrated by A(x)^7 = 1 + 7 x + 28 x^2 + 91 x^3 + 294 x^4 + 1092 x^5 + 5040 x^6 + 29093 x^7 + 203651 x^8 + ... .
A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 206*x^6 + ... = x/series_reversion(x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 120*x^6 + ...).
Related expansions:
log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 + ...;
1 - x/(A(x) - 1) = x + x^2 + 4*x^3 + 21*x^4 + 136*x^5 + 1030*x^6 +...;
(d/dx)((A(x) - 1)/x) = 1 + 4*x + 21*x^2 + 136*x^3 + 1030*x^4 + ... .
MATHEMATICA
a = ConstantArray[0, 20]; a[[1]]=1; a[[2]]=1; a[[3]]=2; Do[a[[n]] = (n-1)*a[[n-1]] + Sum[(j-1)*a[[j]]*a[[n-j]], {j, 2, n-2}], {n, 4, 20}]; Flatten[{1, a}] (* Vaclav Kotesovec after David Callan, Feb 22 2014 *)
InverseSeries[Series[Exp[-x] ExpIntegralEi[x], {x, Infinity, 20}]][[3]] (* Vladimir Reshetnikov, Apr 24 2016 *)
PROG
(PARI) a(n)=if(n<0, 0, if(n<=1, 1, (n-1)*a(n-1)+sum(j=2, n-2, (j-1)*a(j)*a(n-j)); ))
(PARI) a(n)=Vec(x/serreverse(x*Ser(vector(n+1, k, (k-1)!))))[n+1] \\ Paul D. Hanna, Jul 09 2006
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x/(1-x*deriv(A)/A)); polcoeff(A, n)}
(PARI) {a(n)=local(F=1+x*O(x^n)); for(i=0, n, F=1+x*F+x^2*F*deriv(F)+x*O(x^n)); polcoeff(1+x*F, n)} \\ Paul D. Hanna, Sep 02 2008
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2002, Jul 30 2008
EXTENSIONS
More terms from David Wasserman, Jan 26 2005
Minor edits by Vaclav Kotesovec, Aug 01 2015
STATUS
approved