A078468 Distinct compositions of the complete graph with one edge removed (K^-_n).

1, 4, 13, 47, 188, 825, 3937, 20270, 111835, 657423, 4097622, 26965867, 186685725, 1355314108, 10289242825, 81481911259, 671596664012, 5749877335253, 51042081429213, 469037073951694, 4454991580211951, 43677136038927595, 441452153556357966, 4594438326374915007

Offset

0,2

Links

Table of n, a(n) for n=0..23.

A. Knopfmacher and M. E. Mays, Graph Compositions. I: Basic Enumeration, Integers 1(2001), #A04.

Formula

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

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

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

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

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

a(n) = Sum_{k=0..n} Stirling2(n,k) * (k+1)^2. - Ilya Gutkovskiy, Aug 09 2021

Example

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

Maple

with(combinat): a:=n->bell(n+2)-bell(n): seq(a(n), n=0..21); # Zerinvary Lajos, Jul 01 2007

Crossrefs

Cf. A000110, A033452.

Sequence in context: A149440 A149441 A149442 * A354339 A149443 A125656

Adjacent sequences: A078465 A078466 A078467 * A078469 A078470 A078471

Keyword

nonn

Author

Ralf Stephan, Jan 02 2003

Status

approved