A055302 - OEIS

%I #54 Dec 25 2018 22:46:17

%S 1,2,0,6,3,0,24,36,4,0,120,360,140,5,0,720,3600,3000,450,6,0,5040,

%T 37800,54600,18900,1302,7,0,40320,423360,940800,588000,101136,3528,8,

%U 0,362880,5080320,16087680,15876000,5143824,486864,9144,9,0,3628800

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

%C 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 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 313.

%H Alois P. Heinz, <a href="/A055302/b055302.txt">Rows n = 1..141, flattened</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F 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 T(n,k) = (n!/k!)*Stirling2(n-1, n-k). - _Vladeta Jovovic_, Jan 28 2004

%F 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 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 Triangle begins

%e      1,

%e      2,     0;

%e      6,     3,     0;

%e     24,    36,     4,     0;

%e    120,   360,   140,     5,    0;

%e    720,  3600,  3000,   450,    6, 0;

%e   5040, 37800, 54600, 18900, 1302, 7, 0;

%p T:= (n, k)-> (n!/k!)*Stirling2(n-1, n-k):

%p seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Nov 13 2013

%t Table[Table[n!/k! StirlingS2[n-1,n-k], {k,1,n}], {n,0,10}]//Grid  (* _Geoffrey Critzer_, Dec 01 2012 *)

%o (PARI)

%o A055302(n,k)=n!/k!*stirling(n-1, n-k,2);

%o for(n=1,10,for(k=1,n,print1(A055302(n,k),", "));print());

%o \\ _Joerg Arndt_, Oct 27 2014

%Y Row sums give A000169. Columns 1 through 12: A000142, A055303-A055313. Cf. A055314.

%Y Cf. A248120 for a natural refinement.

%K nonn,tabl,eigen

%O 1,2

%A _Christian G. Bower_, May 11 2000