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

0,4

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

Cf. A209881, A091063, A084938.

Cf. A003319.

Sequence in context: A212027 A056543 A357829 * A011800 A112916 A145845

Adjacent sequences: A075831 A075832 A075833 * A075835 A075836 A075837

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