A052863 - OEIS

%I #28 May 13 2022 12:05:26

%S 0,0,2,6,18,65,295,1652,11032,85353,749203,7347384,79564496,942541041,

%T 12121319327,168145213732,2502276609008,39761200642225,

%U 671855234838915,12028625060491336,227451564319791336,4529507975800063337,94751047516476943359,2077192015403191663844

%N Expansion of e.g.f. log(-1/(-1+x))*exp(x) - log(-1/(-1+x)).

%C Previous name was: A simple grammar.

%H Seiichi Manyama, <a href="/A052863/b052863.txt">Table of n, a(n) for n = 0..450</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=831">Encyclopedia of Combinatorial Structures 831</a>

%F E.g.f.: log(-1/(-1+x))*exp(x) - log(-1/(-1+x)).

%F Recurrence: {a(1)=0, a(3)=6, a(2)=2, (-n^3-2*n-3*n^2)*a(n)+(19*n+11*n^2+2*n^3+10)*a(n+1)+(-38*n-12*n^2-n^3-36)*a(n+2)+(41+26*n+4*n^2)*a(n+3)+(-17-5*n)*a(n+4)+2*a(n+5), a(4)=18, a(5)=65}

%F a(n) = A002104(n)-(n-1)!. - _Vladeta Jovovic_, Apr 03 2005

%F a(n) ~ (n-1)! * (exp(1)-1). - _Vaclav Kotesovec_, Sep 29 2013

%F a(n) = Sum_{k=0..n-2} k! * binomial(n,k+1). - _Seiichi Manyama_, May 13 2022

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

%t CoefficientList[Series[Log[-1/(-1+x)]*E^x-Log[-1/(-1+x)], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 29 2013 *)

%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(-log(1-x)*(exp(x)-1)))) \\ _Seiichi Manyama_, May 13 2022

%o (PARI) a(n) = sum(k=0, n-2, k!*binomial(n, k+1)); \\ _Seiichi Manyama_, May 13 2022

%K easy,nonn

%O 0,3

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

%E New name using e.g.f., _Joerg Arndt_, Sep 30 2013