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
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