A052873 E.g.f. A(x) 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
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.
Programs
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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 *) -
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
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
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x) ) ). - Seiichi Manyama, Mar 15 2025
Extensions
New name using e.g.f., Vaclav Kotesovec, Jan 08 2014
Comments