A052845 Expansion of e.g.f.: exp(x^2/(1-x)).

1, 0, 2, 6, 36, 240, 1920, 17640, 183120, 2116800, 26943840, 374220000, 5628934080, 91122071040, 1579034096640, 29155689763200, 571308920582400, 11838533804697600, 258608278645516800, 5938673374272038400, 143003892952893772800, 3602735624977961472000

Offset: 0

Author

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

Keywords

• easy
• nonn

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
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000262, A129652.

Programs

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 *)

PARI

N=33;  x='x+O('x^N);
egf=exp(x^2/(1-x));
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */

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

Extensions

Initial term changed to a(0) = 1, Apr 24 2005