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

1, 2, 6, 26, 148, 1032, 8464, 79592, 842832, 9914336, 128162464, 1804852128, 27489582784, 450089665664, 7880963503872, 146913179393408, 2904309329449216, 60677563647195648, 1335634021282590208, 30891084696208976384, 748854186528315687936

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

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.

Previous name was: A simple grammar.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• T.-X. He, A symbolic operator approach to power series transformation-expansion formulas, JIS 11 (2008) 08.2.7.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 812

Crossrefs

Row sums of A129652.

Cf. A000262.

Programs

Maple

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

Mathematica

CoefficientList[Series[Exp[x/(1 - x)] Exp[x], {x, 0, 20}], x]*
Table[n!, {n, 0, 20}]

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 */

Formula

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

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)}.

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]

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

a(n) ~ exp(2*sqrt(n)-n+1/2)*n^(n-1/4)/sqrt(2). - Vaclav Kotesovec, Oct 09 2012

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

Extensions

New name using e.g.f. from Ilya Gutkovskiy, Aug 13 2021