A052844 - OEIS

%I #43 Jun 02 2024 11:11:17

%S 1,2,6,26,148,1032,8464,79592,842832,9914336,128162464,1804852128,

%T 27489582784,450089665664,7880963503872,146913179393408,

%U 2904309329449216,60677563647195648,1335634021282590208,30891084696208976384,748854186528315687936

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

%C 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 Previous name was: A simple grammar.

%H Vincenzo Librandi, <a href="/A052844/b052844.txt">Table of n, a(n) for n = 0..200</a>

%H 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=812">Encyclopedia of Combinatorial Structures 812</a>

%F E.g.f.: exp(x*(-2+x)/(-1+x)).

%F 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 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 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 a(n) ~ exp(2*sqrt(n)-n+1/2)*n^(n-1/4)/sqrt(2). - _Vaclav Kotesovec_, Oct 09 2012

%F 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 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 CoefficientList[Series[Exp[x/(1 - x)] Exp[x], {x, 0, 20}], x]*

%t Table[n!, {n, 0, 20}]

%o (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 Row sums of A129652.

%Y Cf. A000262.

%K easy,nonn

%O 0,2

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

%E New name using e.g.f. from _Ilya Gutkovskiy_, Aug 13 2021