A052875 E.g.f.: (exp(x)-1)^2/(2-exp(x)).

0, 0, 2, 12, 74, 540, 4682, 47292, 545834, 7087260, 102247562, 1622632572, 28091567594, 526858348380, 10641342970442, 230283190977852, 5315654681981354, 130370767029135900, 3385534663256845322, 92801587319328411132, 2677687796244384203114, 81124824998504073881820

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Stirling transform of A005359(n-1)=[0,0,2,0,24,0,...] is a(n-1)=[0,0,2,12,74,...]. - Michael Somos, Mar 04 2004

Stirling transform of -(-1)^n*A052566(n-1)=[1,-1,4,-6,48,...] is a(n-1)=[1,0,2,12,74,...]. - Michael Somos, Mar 04 2004

Stirling transform of A000142(n)=[0,2,6,24,120,...] is a(n)=[0,2,12,74,...]. - Michael Somos, Mar 04 2004

Stirling transform of A007680(n)=[2,10,42,216,...] is a(n+1)=[2,12,74,...]. - Michael Somos, Mar 04 2004

a(n) is the number of chains in the power set of {1,2,...,n} that do not contain the empty set and do not contain {1,2,...,n}. Equivalently, a(n) is the number of ordered set partitions of {1,2,...,n} into at least 2 classes. - Geoffrey Critzer, Sep 01 2014

Examples

			a(3) = 12 because we have: {{1}}, {{2}}, {{3}}, {{1,2}}, {{1,3}}, {{2,3}}, {{1}, {1,2}}, {{1}, {1,3}}, {{2}, {1,2}}, {{2}, {2,3}}, {{3}, {1,3}}, {{3}, {2,3}}. - _Geoffrey Critzer_, Sep 01 2014
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 846
• R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
• Wikipedia, Ordered Bell number

Crossrefs

Cf. A007047, A038719.

Programs

Maple

spec := [S, {B = Set(Z, 1 <= card), C = Sequence(B, 1 <= card), S=Prod(B, C)}, labeled]: seq(combstruct[count](spec, size=n),  n=0..20);

Mathematica

CoefficientList[Series[(E^x-1)^2/(2-E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 25 2014 *)

PARI

a(n)=if(n<0,0,n!*polcoeff(subst(y^2/(1-y),y,exp(x+x*O(x^n))-1),n))

Sage

def A052875(n):
    return add(add((-1)^(j-i)*binomial(j,i)*i^n for i in range(n+1)) for j in range(n+1)) - 1
[A052875(n) for n in range(19)] # Peter Luschny, Jul 22 2014

Formula

Second column of A084416: Sum_{i=2..n} i!*Stirling2(n, i) = A000670(n)-1. - Vladeta Jovovic, Sep 15 2003

E.g.f.: (exp(x)-1)^2/(2-exp(x)).

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Feb 25 2014

E.g.f.: A(x)*(1/(1 - A(x)) - 1) where A(x)=exp(x)-1. - Geoffrey Critzer, Sep 01 2014

Extensions

New name using e.g.f., Vaclav Kotesovec, Feb 25 2014