A052802 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x))).

1, 1, 5, 47, 660, 12414, 293552, 8374806, 280064600, 10747277832, 465597887592, 22479948822792, 1197060450322800, 69699159437088960, 4405397142701855232, 300408348609092268144, 21983809533066553697280

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 47*x^3/3! + 660*x^4/4! +... [_Paul D. Hanna_, Aug 28 2008]
		

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 760

Crossrefs

Cf. A052819. [From Paul D. Hanna, Aug 28 2008]

Cf. A052894, A198860.

Programs

Maple

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

Mathematica

CoefficientList[1/x*InverseSeries[Series[x + x*Log[1-x], {x, 0, 20}], x],x] * Range[0, 19]! (* Vaclav Kotesovec, Jan 08 2014 *)

Maxima

a(n):=sum(binomial(n+k,n)*k!*sum((-1)^(n+j)/(k-j)!*sum(stirling1(n,j-i)/i!,i,0,j),j,0,k),k,0,n)/(n+1); /* Vladimir Kruchinin, May 09 2013 */

PARI

a(n)=n!*polcoeff((1/x)*serreverse(x+x*log(1-x +x*O(x^n))),n) \\ Paul D. Hanna, Aug 28 2008

Formula

From Paul D. Hanna, Aug 28 2008: (Start)

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

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

E.g.f.: A(x) = (1/x)*Series_Reversion[x + x*log(1-x)]. (End)

a(n)=sum(k=0..n, binomial(n+k,n)*k!*sum(j=0..k, (-1)^(n+j)/(k-j)!*sum(i=0..j, stirling1(n,j-i)/i!)))/(n+1); [Vladimir Kruchinin, May 09 2013]

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

a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * |Stirling1(n,k)|. - Seiichi Manyama, Nov 06 2023

Extensions

New name using e.g.f., Vaclav Kotesovec, Jan 08 2014