login
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.
3

%I #28 Feb 07 2016 21:36:20

%S 1,0,2,3,0,6,4,36,0,24,65,80,360,0,120,306,1950,1200,3600,0,720,4207,

%T 12852,40950,16800,37800,0,5040,38424,235592,359856,764400,235200,

%U 423360,0,40320,573057,2766528,8481312,8636544,13759200,3386880,5080320,0,362880

%N 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 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 T(n,n-1) = n! = A000142(n).

%C Column k = 0 = A060356(n).

%C Row sums = n^(n-1) = A000169(n).

%C 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 Alois P. Heinz, <a href="/A231602/b231602.txt">Rows n = 1..141, flattened</a>

%F E.g.f. satisfies A(x,y) = y*x*A(x,y) + x*( exp(A(x,y)) - A(x,y) ).

%e 1;

%e 0, 2;

%e 3, 0, 6;

%e 4, 36, 0, 24;

%e 65, 80, 360, 0, 120;

%e 306, 1950, 1200, 3600, 0, 720;

%e 4207, 12852, 40950, 16800, 37800, 0, 5040;

%e 38424, 235592, 359856, 764400, 235200, 423360, 0, 40320;

%e ....0..........0........

%e ....|........./ \.......

%e ....0........0...0......

%e .../ \.......|..........

%e ..0 0......0..........

%e 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 with(combinat): C:= binomial:

%p b:= proc(t, i, u) option remember; `if`(t=0, 1,

%p `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)

%p *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))

%p end:

%p T:= (n, k)-> C(n, k)*C(n-1, k)*k! *b(n-1-k$2, n-k):

%p seq(seq(T(n, k), k=0..n-1), n=1..10); # _Alois P. Heinz_, Nov 12 2013

%t 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 Cf. A055302, A206823.

%Y Cf. A248120.

%K nonn,tabl

%O 1,3

%A _Geoffrey Critzer_, Nov 11 2013