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A228550 Triangular array read by rows: T(n,k) is the number of simple labeled graphs with n vertices and k components such that each vertex has even degree; n >= 1, 1 <= k <= n.

%I A228550 #76 Feb 20 2021 17:25:29
%S A228550 1,0,1,1,0,1,3,4,0,1,38,15,10,0,1,720,238,45,20,0,1,26614,5145,868,
%T A228550 105,35,0,1,1858122,215355,21000,2408,210,56,0,1,250586792,16797942,
%U A228550 980371,64260,5628,378,84,0,1,66121926720,2509697144,84370230,3306415,163800,11676,630,120,0,1
%N A228550 Triangular array read by rows: T(n,k) is the number of simple labeled graphs with n vertices and k components such that each vertex has even degree; n >= 1, 1 <= k <= n.
%C A228550 The Bell transform of A033678(n+1). For the definition of the Bell transform, see A264428. - _Peter Luschny_, Jan 17 2016
%H A228550 Alois P. Heinz, <a href="/A228550/b228550.txt">Rows n = 1..45, flattened</a>
%F A228550 E.g.f.: (A(x) + 1)^y, where A(x) = Sum_{n>=1} 2^C(n-1,2) * x^n/n!.
%F A228550 Row sums are 2^binomial(n-1,2) = A006125(n-1).
%F A228550 Column 1 is A033678 (because a connected graph has only one component).
%e A228550 T(3,1) = 1 counts the complete graph on 3 labeled vertices.
%e A228550 T(3,3) = 1 counts the empty graph (no edges) on 3 labeled vertices.
%e A228550 Triangular array T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
%e A228550     1;
%e A228550     0,    1;
%e A228550     1,    0,   1;
%e A228550     3,    4,   0,   1;
%e A228550    38,   15,  10,   0,  1;
%e A228550   720,  238,  45,  20,  0, 1;
%e A228550   ...
%t A228550 nn = 8; e = Sum[2^Binomial[n - 1, 2] x^n/n!, {n, 1, nn}];
%t A228550   Prepend[Drop[Map[Insert[#, 0, -2] &,
%t A228550     Map[Select[#, # > 0 &] &,
%t A228550      Range[0, nn]! CoefficientList[
%t A228550        Series[(e + 1)^y, {x, 0, nn}], {x, y}]]], 2], {1}] // Grid
%o A228550 (Sage) # uses[bell_matrix from A264428]
%o A228550 # Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
%o A228550 bell_matrix(lambda n: A033678(n+1), 9) # _Peter Luschny_, Jan 17 2016
%Y A228550 Cf. A006125, A033678, A264428.
%K A228550 nonn,tabl
%O A228550 1,7
%A A228550 _Geoffrey Critzer_, Aug 27 2013