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A052842 E.g.f. A(x) = series reversion of (1-x)*(1-exp(-x)). 0
0, 1, 3, 23, 290, 5104, 115374, 3185972, 103946688, 3912527016, 166884627360, 7955159511672, 419106982360560, 24182042474691984, 1516563901865906880, 102717031449780063360, 7472238163167018081024 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A simple grammar.

LINKS

Table of n, a(n) for n=0..16.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 809

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

FORMULA

E.g.f. satisfies: A(x) = -log(1 - x/(1-A(x))). [From Encyclopedia of Combinatorial Structures]

a(n) = sum(k=0..n-1, (sum(j=0..k, (sum(i=0..j, (stirling2(i+n-1,j)*C(j,j-i))/ (i+n-1)!))*(-1)^(n+j-1)/(k-j)!))*(n+k-1)!), n>0. - Vladimir Kruchinin, Feb 06 2012

a(n) ~ n^(n-1) * c^n / (sqrt(1+c) * exp(n) * (c-1)^(2*n-1)), where c = LambertW(exp(2)) = 1.5571455989976114... (see A226571). - Vaclav Kotesovec, Jan 08 2014

EXAMPLE

E.g.f.: A(x) = x + 3*x^2/2! + 23*x^3/3! + 290*x^4/4! + 5104*x^5/5! +... which satisfies: A(x) = -log(1 - x/(1-A(x))).

MAPLE

spec := [S, {C=Prod(Z, B), S=Cycle(C), B=Sequence(S)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

MATHEMATICA

CoefficientList[InverseSeries[Series[(-1 + E^(-x))*(x-1), {x, 0, 20}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 08 2014 *)

PROG

(PARI) {a(n)=n!*polcoeff(serreverse((1-exp(-x+O(x^(n+2))))*(1-x)), n)} /* Paul D. Hanna */

(Maxima) a(n):=sum((sum((sum((stirling2(i+n-1, j)*binomial(j, j-i))/(i+n-1)!, i, 0, j))*(-1)^(n+j-1)/(k-j)!, j, 0, k))*(n+k-1)!, k, 0, n-1); /* Vladimir Kruchinin, Feb 06 2012 */

CROSSREFS

Cf. A226571.

Sequence in context: A302117 A006555 A060090 * A307418 A231788 A088692

Adjacent sequences:  A052839 A052840 A052841 * A052843 A052844 A052845

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

Name from a comment by Paul D. Hanna, Jun 22 2011

STATUS

approved

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Last modified February 24 22:57 EST 2020. Contains 332216 sequences. (Running on oeis4.)