A228859 - OEIS

%I #15 Mar 27 2020 13:56:08

%S 1,1,1,3,3,1,19,15,6,1,195,125,45,10,1,3031,1545,480,105,15,1,67263,

%T 27307,7035,1400,210,21,1,2086099,668367,140098,24045,3430,378,28,1,

%U 89224635,22427001,3746925,536214,68355,7434,630,36,1

%N Triangular array read by rows. T(n,k) is the number of labeled bipartite graphs on n nodes having exactly k connected components; n>=1, 1<=k<=n.

%C The Bell transform of A001832(n+1) (without column 0). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 21 2016

%F E.g.f.: sqrt(A(x)^y) where A(x) is the e.g.f. for A047863.

%F Sum_{k=1..n} T(n,k)*2^k = A047863(n).

%e 1,

%e 1, 1,

%e 3, 3, 1,

%e 19, 15, 6, 1,

%e 195, 125, 45, 10, 1,

%e 3031, 1545, 480, 105, 15, 1,

%t nn=9;f[x_]:=Sum[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Drop[Range[0,nn]!CoefficientList[Series[Exp[y Log[f[x]]/2],{x,0,nn}],{x,y}],1]]//Grid

%o (Sage) # uses[bell_matrix from A264428, A001832]

%o # Adds 1,0,0,0,... as column 0 to the triangle.

%o bell_matrix(lambda n: A001832(n+1), 8) # _Peter Luschny_, Jan 21 2016

%Y Row sums are A047864.

%Y Column 1 is A001832.

%Y Cf. A047863.

%K nonn,tabl

%O 1,4

%A _Geoffrey Critzer_, Sep 05 2013