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A231536 Triangular array read by rows. T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} whose functional digraph has exactly k nodes such that no nonrecurrent element is mapped into it. n >= 1, 1 <= k <= n.

%I A231536 #20 Jan 03 2021 16:55:28
%S A231536 1,2,2,6,15,6,24,108,100,24,120,840,1340,705,120,720,7200,17400,15150,
%T A231536 5466,720,5040,68040,231000,296100,171402,46921,5040,40320,705600,
%U A231536 3198720,5644800,4687536,2015272,444648,40320,362880,7983360,46569600,108168480,121144464,73191888,25011576,4625361,362880
%N A231536 Triangular array read by rows.  T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} whose functional digraph has exactly k nodes such that no nonrecurrent element is mapped into it.  n >= 1, 1 <= k <= n.
%F A231536 E.g.f.: 1/(1 - A(x,y)) where A(x,y) is the e.g.f. for A055302.
%e A231536 T(3,3) = 6 because we have: (1,2,3),(2,1,3),(3,2,1),(1,3,2),(2,3,1),(3,1,2).  In these 6 functions represented as a word there are 3 (all) elements with zero nonrecurrent elements mapped to them.
%e A231536 1,
%e A231536 2, 2,
%e A231536 6, 15, 6,
%e A231536 24, 108, 100, 24,
%e A231536 120, 840, 1340, 705, 120,
%e A231536 720, 7200, 17400, 15150, 5466, 720
%t A231536 nn=6;Map[Select[#,#>0&]&,Drop[Range[0,nn]!CoefficientList[Series[1/(1- (-x+x y-ProductLog[-Exp[x (-1+y)] x])),{x,0,nn}],{x,y}],1]]//Grid
%Y A231536 Row sums give: A000312.
%Y A231536 Column k=1 and main diagonal give: A000142.
%K A231536 nonn,tabl
%O A231536 1,2
%A A231536 _Geoffrey Critzer_, Nov 10 2013