OFFSET
0,3
COMMENTS
Turan graph T(2n,n) is also called cocktail party graph, so a(n) is the number of ways to seat n married couples for one or a few tables in such a manner that no table is fully occupied by any couple.
If we dissect (n-1)-skeleton of n-cube along some (n-2)-edges into some parts, then a(n) is the number of ways of such dissections.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Turan Graph
FORMULA
EXAMPLE
For n=1, Turan graph T(2,1) (2-empty graph) shall be partitioned into two singleton subgraphs (1 way), a(1)=1.
For n=2, Turan graph T(4,2) (square graph) shall be partitioned into: the same square graph (1 way) or one singleton + one 3-path subgraphs (4 ways) or two singleton + one 2-path subgraphs (4 ways) or two 2-path subgraphs (2 ways) or four singleton subgraphs (1 way), a(2)=12.
MAPLE
A282010 := proc(n)
add((-1)^(n-j)*combinat[bell](2*j)*binomial(n, j), j=0..n) ;
end proc:
seq(A282010(n), n=0..20) ; # R. J. Mathar, Jun 27 2024
MATHEMATICA
a[n_]:=BellB[2n]; Table[Sum[((-1)^(n-j))*a[j]*Binomial[n, j], {j, 0, n}], {n, 0, 17}] (* Indranil Ghosh, Feb 25 2017 *)
PROG
(PARI) bell(n) = polcoeff( sum( k=0, n, prod(i=1, k, x/(1 - i*x)), x^n * O(x)), n)
a(n) = sum(j=0, n, ((-1)^(n-j))*bell(2*j)*binomial(n, j)); \\ Michel Marcus, Feb 05 2017
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Tengiz Gogoberidze, Feb 04 2017
EXTENSIONS
More terms from Michel Marcus, Feb 05 2017
STATUS
approved