A052851 Expansion of e.g.f. 1/2 - (1/2)*(1+4*log(1-x))^(1/2).

0, 1, 3, 20, 220, 3424, 69008, 1706256, 49956240, 1689497376, 64799254752, 2778906776832, 131756614920192, 6843405231815424, 386414606189283072, 23567401521343170048, 1543994621969805135360, 108137637714495023354880, 8062825821198926369725440

Offset

0,3

Comments

Previous name was: A simple grammar.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..360

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 819

Index entries for reversions of series

Formula

E.g.f.: 1/2 - (1/2)*(1-4*log(-1/(-1+x)))^(1/2).

a(n) = Sum_{k=1..n} Stirling1(n,k)*k!*C(2*k-2,k-1)/k*(-1)^(n+k). - Vladimir Kruchinin, May 12 2012

a(n) ~ n^(n-1)/(sqrt(2)*exp(3*n/4)*(exp(1/4)-1)^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013

From Seiichi Manyama, Sep 09 2024: (Start)

E.g.f. satisfies A(x) = (-log(1 - x)) / (1 - A(x)).

E.g.f.: Series_Reversion( 1 - exp(-x * (1 - x)) ). (End)

Maple

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

Mathematica

CoefficientList[Series[1/2-1/2*(1+4*Log[1-x])^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

Prog

(Maxima) a(n):=sum(stirling1(n, k)*k!*binomial(2*k-2, k-1)/k*(-1)^(n+k), k, 1, n); /* Vladimir Kruchinin, May 12 2012 */

Crossrefs

Cf. A048287, A052886, A087138.

Cf. A371314, A371315.

Cf. A371327, A376041, A376042.

Sequence in context: A113333 A206405 A307363 * A262233 A058477 A119758

Adjacent sequences: A052848 A052849 A052850 * A052852 A052853 A052854

Keyword

easy,nonn

Author

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

Extensions

New name using e.g.f., Vaclav Kotesovec, Sep 30 2013

Status

approved