A078468 - OEIS

%I #34 Aug 17 2021 19:30:37

%S 1,4,13,47,188,825,3937,20270,111835,657423,4097622,26965867,

%T 186685725,1355314108,10289242825,81481911259,671596664012,

%U 5749877335253,51042081429213,469037073951694,4454991580211951,43677136038927595,441452153556357966,4594438326374915007

%N Distinct compositions of the complete graph with one edge removed (K^-_n).

%H A. Knopfmacher and M. E. Mays, <a href="http://www.emis.de/journals/INTEGERS/papers/b4/b4.Abstract.html">Graph Compositions. I: Basic Enumeration</a>, Integers 1(2001), #A04.

%F a(n) = A000110(n+2) - A000110(n).

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

%F G.f.: (G(0)*(1-x)-1-x)/x^2 where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 03 2013

%F G.f.: - G(0)*(1+1/x) where G(k) = 1 - 1/(1-x*(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Feb 07 2013

%F G.f.: (Q(0) -1)*(1+x)/x^2, where Q(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*(k+1))*(1-x*(k+2))/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 10 2013

%F a(n) = Sum_{k=0..n} Stirling2(n,k) * (k+1)^2. - _Ilya Gutkovskiy_, Aug 09 2021

%e a(5) = A000110(7)-A000110(5) = 825.

%p with(combinat): a:=n->bell(n+2)-bell(n): seq(a(n), n=0..21); # _Zerinvary Lajos_, Jul 01 2007

%Y Cf. A000110, A033452.

%K nonn

%O 0,2

%A _Ralf Stephan_, Jan 02 2003