A052845 - OEIS

%I #50 Feb 22 2022 09:49:18

%S 1,0,2,6,36,240,1920,17640,183120,2116800,26943840,374220000,

%T 5628934080,91122071040,1579034096640,29155689763200,571308920582400,

%U 11838533804697600,258608278645516800,5938673374272038400,143003892952893772800,3602735624977961472000

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

%C 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

%H Harvey P. Dale, <a href="/A052845/b052845.txt">Table of n, a(n) for n = 0..400</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=813">Encyclopedia of Combinatorial Structures 813</a>

%H R. A. Proctor, <a href="https://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F 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.

%F Inverse binomial transform of A000262: Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A000262(k). - _Vladeta Jovovic_, _Vladimir Baltic_, Oct 29 2002

%F 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

%F 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

%F E.g.f.: Product_{k>1} exp(x^k). - _Seiichi Manyama_, Sep 29 2017

%F a(0) = 1; a(n) = Sum_{k=2..n} binomial(n-1,k-1) * k! * a(n-k). - _Ilya Gutkovskiy_, Feb 09 2020

%F a(n) = Sum_{k=0..n} (-1)^k * A129652(n,k). - _Alois P. Heinz_, Feb 21 2022

%p 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);

%t With[{nn=20},CoefficientList[Series[Exp[x^2/(1-x)],{x,0,nn}], x] Range[ 0,nn]!] (* _Harvey P. Dale_, May 31 2012 *)

%o (PARI)

%o N=33;  x='x+O('x^N);

%o egf=exp(x^2/(1-x));

%o Vec(serlaplace(egf))

%o /* _Joerg Arndt_, Sep 15 2012 */

%Y Cf. A000262, A129652.

%K easy,nonn

%O 0,3

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

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