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A231602 Triangular array read by rows: T(n,k) is the number of rooted labeled trees on n nodes that have exactly k nodes with outdegree = 1, n>=1, 0<=k<=n-1.

%I A231602 #28 Feb 07 2016 21:36:20
%S A231602 1,0,2,3,0,6,4,36,0,24,65,80,360,0,120,306,1950,1200,3600,0,720,4207,
%T A231602 12852,40950,16800,37800,0,5040,38424,235592,359856,764400,235200,
%U A231602 423360,0,40320,573057,2766528,8481312,8636544,13759200,3386880,5080320,0,362880
%N A231602 Triangular array read by rows: T(n,k) is the number of rooted labeled trees on n nodes that have exactly k nodes with outdegree = 1, n>=1, 0<=k<=n-1.
%C A231602 T(n,k) is also the number of functions f:{1,2,...,n-1}->{1,2,...,n} that have exactly k elements whose preimage has cardinality = 1.
%C A231602 T(n,n-1) = n! = A000142(n).
%C A231602 Column k = 0 = A060356(n).
%C A231602 Row sums = n^(n-1) = A000169(n).
%C A231602 Refinement given by A248120. Sum coefficients of the partition polynomials with h_1 = (1') = t and all other h_n = (n') = 1 to obtain this entry. - _Tom Copeland_, Feb 01 2016
%H A231602 Alois P. Heinz, <a href="/A231602/b231602.txt">Rows n = 1..141, flattened</a>
%F A231602 E.g.f. satisfies A(x,y) = y*x*A(x,y) + x*( exp(A(x,y)) - A(x,y) ).
%e A231602 1;
%e A231602 0, 2;
%e A231602 3, 0, 6;
%e A231602 4, 36, 0, 24;
%e A231602 65, 80, 360, 0, 120;
%e A231602 306, 1950, 1200, 3600, 0, 720;
%e A231602 4207, 12852, 40950, 16800, 37800, 0, 5040;
%e A231602 38424, 235592, 359856, 764400, 235200, 423360, 0, 40320;
%e A231602 ....0..........0........
%e A231602 ....|........./ \.......
%e A231602 ....0........0...0......
%e A231602 .../ \.......|..........
%e A231602 ..0   0......0..........
%e A231602 T(4,1) = 36.  Both of these graphs on 4 nodes have exactly 1 node that has outdegree = 1.  There are 12 + 24 = 36 labelings.
%p A231602 with(combinat): C:= binomial:
%p A231602 b:= proc(t, i, u) option remember; `if`(t=0, 1,
%p A231602       `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
%p A231602       *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
%p A231602     end:
%p A231602 T:= (n, k)-> C(n, k)*C(n-1, k)*k! *b(n-1-k$2, n-k):
%p A231602 seq(seq(T(n, k), k=0..n-1), n=1..10);  # _Alois P. Heinz_, Nov 12 2013
%t A231602 nn=8;Table[Table[Drop[Range[0,nn]!CoefficientList[Series[-ProductLog[x/(-1-x+x y)],{x,0,nn}],{x,y}],1][[r,c]],{c,1,r}],{r,1,nn}]//Grid
%Y A231602 Cf. A055302, A206823.
%Y A231602 Cf. A248120.
%K A231602 nonn,tabl
%O A231602 1,3
%A A231602 _Geoffrey Critzer_, Nov 11 2013