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A247232 Triangular array read by rows: T(n,k) is the number of pre-orders on an n-set with exactly k connected components in its digraph representation, n>=1, 1<=k<=n.

%I A247232 #31 Mar 25 2020 14:46:36
%S A247232 1,3,1,19,9,1,233,103,18,1,4851,1735,325,30,1,158175,43201,7320,785,
%T A247232 45,1,7724333,1567783,218491,22960,1610,63,1,550898367,82142943,
%U A247232 8856974,818461,59570,2954,84,1,56536880923,6187176225,496368181,37205658,2518131,135198,4998,108,1
%N A247232 Triangular array read by rows: T(n,k) is the number of pre-orders on an n-set with exactly k connected components in its digraph representation, n>=1, 1<=k<=n.
%C A247232 The Bell transform of A001929(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%F A247232 E.g.f.: A(exp(x)-1)^y where A(x) is the e.g.f. for A001035.
%e A247232 1;
%e A247232 3,         1;
%e A247232 19,        9,        1;
%e A247232 233,       103,      18,      1;
%e A247232 4851,      1735,     325,     30,     1;
%e A247232 158175,    43201,    7320,    785,    45,    1;
%e A247232 7724333,   1567783,  218491,  22960,  1610,  63,   1;
%e A247232 550898367, 82142943, 8856974, 818461, 59570, 2954, 84, 1;
%t A247232 A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]];
%t A247232 lg = Length[A001035];
%t A247232 A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
%t A247232 Rest[CoefficientList[#, y]]& /@ (CoefficientList[A[Exp[x] - 1]^y + O[x]^lg, x]*Range[0, lg - 1]!) // Flatten (* _Jean-François Alcover_, Jan 01 2020 *)
%o A247232 (Sage) # uses[bell_matrix from A264428]
%o A247232 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A247232 topo = oeis('A001929')  # Fetch the data via Internet.
%o A247232 A001929List = topo.first_terms()
%o A247232 A001929 = lambda n: A001929List[n]
%o A247232 bell_matrix(lambda n: A001929(n+1), 10) # _Peter Luschny_, Jan 18 2016, updated Mar 25 2020
%Y A247232 Column 1 = A001929.
%Y A247232 Row sums = A000798.
%K A247232 nonn,tabl
%O A247232 1,2
%A A247232 _Geoffrey Critzer_, Nov 27 2014