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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. 1

%I

%S 1,3,1,11,9,1,58,71,18,1,409,620,245,30,1,3606,6274,3255,625,45,1,

%T 38149,73339,45724,11795,1330,63,1,470856,977780,697004,221529,33880,

%U 2506,84,1,6641793,14678712,11602394,4309956,823179,82908,4326,108,1

%N 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 The Bell transform of (-1)^n*A009444(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016

%H Alois P. Heinz, <a href="/A228534/b228534.txt">Rows n = 1..90, flattened</a>

%F E.g.f.: 1/(1 - x*exp(x))^y.

%e 1;

%e 3, 1;

%e 11, 9, 1;

%e 58, 71, 18, 1;

%e 409, 620, 245, 30, 1;

%e 3606, 6274, 3255, 625, 45, 1;

%e 38149, 73339, 45724, 11795, 1330, 63, 1;

%e 470856, 977780, 697004, 221529, 33880, 2506, 84, 1;

%p # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p g := n -> add(m^(n-m)*m!*binomial(n+1,m), m=1..n+1);

%p BellMatrix(g, 9); # _Peter Luschny_, Jan 29 2016

%t nn = 8; a = x Exp[x];

%t Map[Select[#, # > 0 &] &,

%t Drop[Range[0, nn]! CoefficientList[

%t Series[1/(1 - a)^y, {x, 0, nn}], {x, y}], 1]] // Grid

%t (* Second program: *)

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[Function[n, (n+1)! Sum[m^(n-m)/(n-m+1)!, {m, 1, n+1}]], rows = 12];

%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-Fran├žois Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%o (Sage)

%o # The function bell_matrix is defined in A264428.

%o # Adds a column 1,0,0,0, ... at the left side of the triangle.

%o bell_matrix(lambda n: (-1)^n*A009444(n+1), 10) # _Peter Luschny_, Jan 18 2016

%Y Row sums = A006153.

%Y Column 1 = |A009444|.

%Y Cf. A199673.

%K nonn,tabl

%O 1,2

%A _Geoffrey Critzer_, Aug 24 2013

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Last modified October 21 18:53 EDT 2019. Contains 328308 sequences. (Running on oeis4.)