cp's OEIS Frontend

A282010 Number of ways to partition Turan graph T(2n,n) into connected components.

%I A282010 #19 Feb 16 2025 08:33:40
%S A282010 1,1,12,163,3411,97164,3576001,163701521,9064712524,594288068019,
%T A282010 45352945127123,3973596101084108,395147058261233761,
%U A282010 44170986458602383553,5504694207040057913164,759355292729159336345955,115228949414563130433140659,19129024114529146183236435660
%N A282010 Number of ways to partition Turan graph T(2n,n) into connected components.
%C A282010 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.
%C A282010 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.
%H A282010 Indranil Ghosh, <a href="/A282010/b282010.txt">Table of n, a(n) for n = 0..100</a>
%H A282010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H A282010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TuranGraph.html">Turan Graph</a>
%F A282010 a(n) = Sum_{j=0..n} ((-1)^(n-j))*A020557(j)*binomial(n,j).
%F A282010 a(n) = Sum_{j=0..n} ((-1)^(n-j))*A000110(2*j)*binomial(n,j).
%e A282010 For n=1, Turan graph T(2,1) (2-empty graph) shall be partitioned into two singleton subgraphs (1 way), a(1)=1.
%e A282010 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.
%p A282010 A282010 := proc(n)
%p A282010     add((-1)^(n-j)*combinat[bell](2*j)*binomial(n,j),j=0..n) ;
%p A282010 end proc:
%p A282010 seq(A282010(n),n=0..20) ; # _R. J. Mathar_, Jun 27 2024
%t A282010 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 *)
%o A282010 (PARI) bell(n) = polcoeff( sum( k=0, n, prod(i=1, k, x/(1 - i*x)), x^n * O(x)), n)
%o A282010 a(n) = sum(j=0, n, ((-1)^(n-j))*bell(2*j)*binomial(n,j)); \\ _Michel Marcus_, Feb 05 2017
%Y A282010 Cf. A000110, A020557
%K A282010 nonn,easy
%O A282010 0,3
%A A282010 _Tengiz Gogoberidze_, Feb 04 2017
%E A282010 More terms from _Michel Marcus_, Feb 05 2017