|
|
A052875
|
|
E.g.f.: (exp(x)-1)^2/(2-exp(x)).
|
|
10
|
|
|
0, 0, 2, 12, 74, 540, 4682, 47292, 545834, 7087260, 102247562, 1622632572, 28091567594, 526858348380, 10641342970442, 230283190977852, 5315654681981354, 130370767029135900, 3385534663256845322, 92801587319328411132, 2677687796244384203114, 81124824998504073881820
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
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
|
|
LINKS
|
|
|
FORMULA
|
E.g.f.: (exp(x)-1)^2/(2-exp(x)).
E.g.f.: A(x)*(1/(1 - A(x)) - 1) where A(x)=exp(x)-1. - Geoffrey Critzer, Sep 01 2014
|
|
EXAMPLE
|
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
|
|
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 *)
|
|
PROG
|
(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst(y^2/(1-y), y, exp(x+x*O(x^n))-1), n))
(Sage)
return add(add((-1)^(j-i)*binomial(j, i)*i^n for i in range(n+1)) for j in range(n+1)) - 1
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|