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A075834 Coefficients of power series A(x) such that n-th term of A(x)^n = n! x^(n-1) for n>0. 19
1, 1, 1, 2, 7, 34, 206, 1476, 12123, 111866, 1143554, 12816572, 156217782, 2057246164, 29111150620, 440565923336, 7101696260883, 121489909224618, 2198572792193786, 41966290373704332, 842706170872913634 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also, number of stablized-interval-free permutations on [n] (see Callan link).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

F. Ardila, F. Rincón, L. Williams, Positroids and non-crossing partitions, arXiv preprint arXiv:1308.2698, 2013

David Callan, Counting stabilized-interval-free permutations, arXiv:math/0310157 [math.CO]

David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.

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 (from 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). [From Paul D. Hanna, Sep 02 2008]

a(n) ~ exp(-1) * n!. - 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 7-th 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 +..;

derivative[(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 *)

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)} [From Paul D. Hanna, Sep 02 2008]

CROSSREFS

Cf. A209881, A091063, A084938.

Cf. A003319.

Sequence in context: A145345 A212027 A056543 * 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

STATUS

approved

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Last modified November 23 22:48 EST 2014. Contains 249866 sequences.