OFFSET
0,3
COMMENTS
Number of partitions of {1,..,n} into any number of lists of size >1, where a list means an ordered subset, cf. A000262. - Vladeta Jovovic, Vladimir Baltic, Oct 29 2002
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 813
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
N. J. A. Sloane, Transforms
FORMULA
D-finite with recurrence: a(0)=1, a(1)=0, a(2)=2, (n^2+3*n+2)*a(n)+(n^2+n-2)*a(n+1)+(-4-2*n)*a(n+2)+a(n+3)=0.
Inverse binomial transform of A000262: Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A000262(k). - Vladeta Jovovic, Vladimir Baltic, Oct 29 2002
a(n) ~ n^(n-1/4)*exp(-3/2+2*sqrt(n)-n)/sqrt(2) * (1 + 43/(48*sqrt(n))). - Vaclav Kotesovec, Jun 24 2013, extended Dec 01 2021
E.g.f.: E(0) - 1, where E(k) = 2 + x^2/((2*k+1)*(1-x) - x^2/E(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Dec 30 2013
E.g.f.: Product_{k>1} exp(x^k). - Seiichi Manyama, Sep 29 2017
a(0) = 1; a(n) = Sum_{k=2..n} binomial(n-1,k-1) * k! * a(n-k). - Ilya Gutkovskiy, Feb 09 2020
a(n) = Sum_{k=0..n} (-1)^k * A129652(n,k). - Alois P. Heinz, Feb 21 2022
MAPLE
spec := [S, {B=Sequence(Z, 1 <= card), C=Prod(Z, B), S= Set(C, 1 <= card)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
With[{nn=20}, CoefficientList[Series[Exp[x^2/(1-x)], {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, May 31 2012 *)
PROG
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Initial term changed to a(0) = 1, Apr 24 2005
STATUS
approved