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A055302 Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n.

%I A055302 #54 Dec 25 2018 22:46:17
%S A055302 1,2,0,6,3,0,24,36,4,0,120,360,140,5,0,720,3600,3000,450,6,0,5040,
%T A055302 37800,54600,18900,1302,7,0,40320,423360,940800,588000,101136,3528,8,
%U A055302 0,362880,5080320,16087680,15876000,5143824,486864,9144,9,0,3628800
%N A055302 Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n.
%C A055302 Beginning with the second row, dividing each row by n gives the mirror of row n-1 of A141618. Under the exponential transform, the mirror of A141618 is generated, relating the number of connected graphs here to the number of disconnected graphs associated with A141618 (cf. A127671 and A036040). - _Tom Copeland_, Oct 25 2014
%D A055302 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 313.
%H A055302 Alois P. Heinz, <a href="/A055302/b055302.txt">Rows n = 1..141, flattened</a>
%H A055302 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A055302 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A055302 E.g.f. (relative to x) satisfies: A(x,y) = xy + x*exp(A(x,y)) - x. Divides by n and shifts up under exponential transform.
%F A055302 T(n,k) = (n!/k!)*Stirling2(n-1, n-k). - _Vladeta Jovovic_, Jan 28 2004
%F A055302 T(n,k) = A055314(n,k)*(n-k) + A055314(n,k+1)*(k+1). The first term is the number of such trees with root degree > 1 while the second term is the number of such trees with root degree = 1. This simplifies to the above formula by Vladeta Jovovic. - _Geoffrey Critzer_, Dec 01 2012
%F A055302 E.g.f.: G(x,t) = log[1 + t * N(x*t,1/t)], where N(x,t) is the e.g.f. of A141618. Also, G(x*t,1/t)= log[1 + N(x,t)/t] is the comp. inverse in x of x / [1 + t * (e^x - 1)]. - _Tom Copeland_, Oct 26 2014
%e A055302 Triangle begins
%e A055302      1,
%e A055302      2,     0;
%e A055302      6,     3,     0;
%e A055302     24,    36,     4,     0;
%e A055302    120,   360,   140,     5,    0;
%e A055302    720,  3600,  3000,   450,    6, 0;
%e A055302   5040, 37800, 54600, 18900, 1302, 7, 0;
%p A055302 T:= (n, k)-> (n!/k!)*Stirling2(n-1, n-k):
%p A055302 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Nov 13 2013
%t A055302 Table[Table[n!/k! StirlingS2[n-1,n-k], {k,1,n}], {n,0,10}]//Grid  (* _Geoffrey Critzer_, Dec 01 2012 *)
%o A055302 (PARI)
%o A055302 A055302(n,k)=n!/k!*stirling(n-1, n-k,2);
%o A055302 for(n=1,10,for(k=1,n,print1(A055302(n,k),", "));print());
%o A055302 \\ _Joerg Arndt_, Oct 27 2014
%Y A055302 Row sums give A000169. Columns 1 through 12: A000142, A055303-A055313. Cf. A055314.
%Y A055302 Cf. A248120 for a natural refinement.
%K A055302 nonn,tabl,eigen
%O A055302 1,2
%A A055302 _Christian G. Bower_, May 11 2000