A052873 E.g.f. satisfies A(x) = exp(x*A(x)/(1 - x*A(x))).

1, 1, 5, 46, 629, 11496, 263857, 7301680, 236748969, 8806142080, 369714769181, 17296339048704, 892335712777885, 50333180563864576, 3081739132775658825, 203555129140352505856, 14428195498061848405073, 1092403962489972428144640, 87990832863810814525250869

Offset

0,3

Comments

Previous name was: A simple grammar.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..357

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 844.

R. Lorentz, S. Tringali, C. H. Yan, Generalized Goncarov polynomials, arXiv preprint arXiv:1511.04039 [math.CO], 2015.

Formula

E.g.f.: exp(RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z)).

1 = Sum_{n>=0} a(n)*exp((n+1)*x/(x-1))*x^n/n!. - Vladeta Jovovic, Jul 20 2005

a(n) = Sum_{k=0..n} (n+1)^(k-1)*n!/k!*binomial(n-1,k-1). - Vladeta Jovovic, Jul 02 2006

E.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! / (1-x*A(x))^n. - Paul D. Hanna, Sep 08 2012

Equivalently:

E.g.f. satisfies: A(x) = exp( x*A(x)/(1 - x*A(x)) ). - Olivier Gérard, Dec 29 2013

a(n) ~ (sqrt(5)-1) * 2^(n-1/2) * n^(n-1) * exp((sqrt(5)-1 + (sqrt(5)-3)*n)/2) / (5^(1/4) * (3-sqrt(5))^(n+1/2)). - Vaclav Kotesovec, Jan 08 2014

a(n) = n!*hypergeom([1-n],[2],-n-1) for n >= 1. - Peter Luschny, Apr 20 2016

Maple

spec := [S, {C=Sequence(B, 1 <= card), S=Set(C), B=Prod(Z, S)}, labeled]:
seq(combstruct[count](spec, size=n), n=0..20);
# Alternatively:
a := n -> `if`(n=0, 1, n!*hypergeom([1-n], [2], -n-1)):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Apr 20 2016

Mathematica

Table[Sum[(n+1)^(k-1)*n!/k!*Binomial[n-1, k-1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 08 2014 *)

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n, (n+1)^(k-1)*n!/k!*binomial(n-1, k-1)))} \\ Paul D. Hanna, Sep 08 2012
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (m+1)^(m-1)*x^m/m!/(1-x*A+x*O(x^n))^m)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Sep 08 2012

Crossrefs

Cf. A052868, A088690, A161630.

Sequence in context: A367154 A121631 A071214 * A052894 A363355 A292408

Adjacent sequences: A052870 A052871 A052872 * A052874 A052875 A052876

Keyword

easy,nonn

Author

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

Extensions

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

Status

approved