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A189898 Triangular array read by rows. T(n,k) is the number of digraphs with n labeled nodes having exactly k undirected (or weak) components, n >= 1, 1 <= k <= n.

%I A189898 #34 Jan 17 2022 14:02:43
%S A189898 1,3,1,54,9,1,3834,243,18,1,1027080,20790,675,30,1,1067308488,6364170,
%T A189898 67635,1485,45,1,4390480193904,7543111716,23031540,171045,2835,63,1,
%U A189898 72022346388181584,35217115838604,30469951764,63580545,370440,4914,84,1
%N A189898 Triangular array read by rows.  T(n,k) is the number of digraphs with n labeled nodes having exactly k undirected (or weak) components, n >= 1, 1 <= k <= n.
%C A189898 The Bell transform of A003027(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%H A189898 Alois P. Heinz, <a href="/A189898/b189898.txt">Rows n = 1..32, flattened</a>
%F A189898 E.g.f. for column k: log(A(x))^k/k! where A(x) is the e.g.f. for A053763.
%e A189898 1
%e A189898 3       1
%e A189898 54      9     1
%e A189898 3834    243   18   1
%e A189898 1027080 20790 675  30  1
%p A189898 T:= (n, k)-> coeff(series(log(add(2^(i^2-i) *x^i/i!, i=0..n))^k /k!,
%p A189898                    x, n+1), x, n) *n!:
%p A189898 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, May 01 2011
%t A189898 a= Sum[4^Binomial[n,2]x^n/n!,{n,0,10}];
%t A189898 Transpose[Map[Drop[#, 1] &,Table[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}], x], {n, 1, 10}]]] // Grid
%o A189898 (Sage) # uses[bell_matrix from A264428, A003027]
%o A189898 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A189898 bell_matrix(lambda n: A003027(n+1), 10) # _Peter Luschny_, Jan 18 2016
%Y A189898 Column 1 = A003027, row sums = A053763, lower diagonal = A045943.
%K A189898 nonn,tabl
%O A189898 1,2
%A A189898 _Geoffrey Critzer_, May 01 2011
%E A189898 Name clarified by _Andrew Howroyd_, Jan 11 2022