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A052844 E.g.f.: exp(x*(2-x)/(1-x)).

%I A052844 #43 Jun 02 2024 11:11:17
%S A052844 1,2,6,26,148,1032,8464,79592,842832,9914336,128162464,1804852128,
%T A052844 27489582784,450089665664,7880963503872,146913179393408,
%U A052844 2904309329449216,60677563647195648,1335634021282590208,30891084696208976384,748854186528315687936
%N A052844 E.g.f.: exp(x*(2-x)/(1-x)).
%C A052844 An unspecified number of sign-in sheets are available at a meeting of n people.  The attendees sign in on one of the sheets in the order that they arrive at the meeting.  But some, none, or all of the attendees forget to sign in.  a(n) is the number of ways this can happen.
%C A052844 Previous name was: A simple grammar.
%H A052844 Vincenzo Librandi, <a href="/A052844/b052844.txt">Table of n, a(n) for n = 0..200</a>
%H A052844 T.-X. He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/He/he51.html">A symbolic operator approach to power series transformation-expansion formulas</a>, JIS 11 (2008) 08.2.7.
%H A052844 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=812">Encyclopedia of Combinatorial Structures 812</a>
%F A052844 E.g.f.: exp(x*(-2+x)/(-1+x)).
%F A052844 Recurrence: {a(0)=1, a(1)=2, a(2)=6, (-2-n^2-3*n)*a(n)+(n^2+5*n+6)*a(n+1)+(-2*n-6)*a(n+2)+a(n+3)}.
%F A052844 a(n) = n!*sum(m=1,n, ((sum(j=0,m, binomial(m,j)*binomial(n-j-1,m-j-1))))/m!)+1; [_Vladimir Kruchinin_, May 02 2012]
%F A052844 E.g.f. = exp(x)*exp(x/(1-x)) so a(n) = Sum_{k = 0..n} binomial(n,k)*A000262(k). - _Peter Bala_ May 14 2012
%F A052844 a(n) ~ exp(2*sqrt(n)-n+1/2)*n^(n-1/4)/sqrt(2). - _Vaclav Kotesovec_, Oct 09 2012
%F A052844 a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * k! * a(n-k). - _Ilya Gutkovskiy_, Aug 13 2021
%p A052844 spec := [S,{B=Sequence(Z,1 <= card),C=Union(Z,B),S=Set(C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052844 CoefficientList[Series[Exp[x/(1 - x)] Exp[x], {x, 0, 20}], x]*
%t A052844 Table[n!, {n, 0, 20}]
%o A052844 (Maxima) a(n):=n!*sum(((sum(binomial(m,j)*binomial(n-j-1,m-j-1),j,0,m)))/m!,m,1,n)+1; /* _Vladimir Kruchinin_, May 02 2012 */
%Y A052844 Row sums of A129652.
%Y A052844 Cf. A000262.
%K A052844 easy,nonn
%O A052844 0,2
%A A052844 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052844 New name using e.g.f. from _Ilya Gutkovskiy_, Aug 13 2021