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A228534 Triangular array read by rows: T(n,k) is the number of functional digraphs on {1,2,...,n} such that every element is mapped to a recurrent element and there are exactly k cycles, n>=1, 1<=k<=n.

%I A228534 #27 Mar 27 2020 13:55:19
%S A228534 1,3,1,11,9,1,58,71,18,1,409,620,245,30,1,3606,6274,3255,625,45,1,
%T A228534 38149,73339,45724,11795,1330,63,1,470856,977780,697004,221529,33880,
%U A228534 2506,84,1,6641793,14678712,11602394,4309956,823179,82908,4326,108,1
%N A228534 Triangular array read by rows: T(n,k) is the number of functional digraphs on {1,2,...,n} such that every element is mapped to a recurrent element and there are exactly k cycles, n>=1, 1<=k<=n.
%C A228534 The Bell transform of (-1)^n*A009444(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%H A228534 Alois P. Heinz, <a href="/A228534/b228534.txt">Rows n = 1..90, flattened</a>
%F A228534 E.g.f.: 1/(1 - x*exp(x))^y.
%e A228534        1;
%e A228534        3,      1;
%e A228534       11,      9,      1;
%e A228534       58,     71,     18,      1;
%e A228534      409,    620,    245,     30,     1;
%e A228534     3606,   6274,   3255,    625,    45,    1;
%e A228534    38149,  73339,  45724,  11795,  1330,   63,  1;
%e A228534   470856, 977780, 697004, 221529, 33880, 2506, 84, 1;
%p A228534 # The function BellMatrix is defined in A264428.
%p A228534 # Adds (1,0,0,0, ..) as column 0.
%p A228534 g := n -> add(m^(n-m)*m!*binomial(n+1,m), m=1..n+1);
%p A228534 BellMatrix(g, 9); # _Peter Luschny_, Jan 29 2016
%t A228534 nn = 8; a = x Exp[x];
%t A228534 Map[Select[#, # > 0 &] &,
%t A228534   Drop[Range[0, nn]! CoefficientList[
%t A228534      Series[1/(1 - a)^y, {x, 0, nn}], {x, y}], 1]] // Grid
%t A228534 (* Second program: *)
%t A228534 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A228534 B = BellMatrix[Function[n, (n+1)! Sum[m^(n-m)/(n-m+1)!, {m, 1, n+1}]], rows = 12];
%t A228534 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A228534 (Sage) # uses[bell_matrix from A264428, A009444]
%o A228534 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A228534 bell_matrix(lambda n: (-1)^n*A009444(n+1), 10) # _Peter Luschny_, Jan 18 2016
%Y A228534 Row sums = A006153.
%Y A228534 Column 1 = |A009444|.
%Y A228534 Cf. A199673.
%K A228534 nonn,tabl
%O A228534 1,2
%A A228534 _Geoffrey Critzer_, Aug 24 2013