A075834 Coefficients of power series A(x) such that n-th term of A(x)^n = n! x^(n-1) for n > 0.

1, 1, 1, 2, 7, 34, 206, 1476, 12123, 111866, 1143554, 12816572, 156217782, 2057246164, 29111150620, 440565923336, 7101696260883, 121489909224618, 2198572792193786, 41966290373704332, 842706170872913634, 17759399688526009020, 391929722837419044420

Offset: 0

Author

Paul D. Hanna, Oct 14 2002, Jul 30 2008

Keywords

• nonn

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

Examples

			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 + ... .
		

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.

Crossrefs

Cf. A209881, A091063, A084938.

Cf. A003319.

Programs

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 *)

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

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

Extensions

More terms from David Wasserman, Jan 26 2005

Minor edits by Vaclav Kotesovec, Aug 01 2015