A244372 - OEIS

A244372

Number T(n,k) of unlabeled rooted trees with n nodes and maximal outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

%I #23 Sep 07 2018 17:22:07

%S 1,0,1,0,1,1,0,1,2,1,0,1,5,2,1,0,1,10,6,2,1,0,1,22,16,6,2,1,0,1,45,43,

%T 17,6,2,1,0,1,97,113,49,17,6,2,1,0,1,206,300,136,50,17,6,2,1,0,1,450,

%U 787,386,142,50,17,6,2,1,0,1,982,2074,1081,409,143,50,17,6,2,1

%N Number T(n,k) of unlabeled rooted trees with n nodes and maximal outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

%H Alois P. Heinz, <a href="/A244372/b244372.txt">Rows n = 1..141, flattened</a>

%e The A000081(5) = 9 rooted trees with 5 nodes sorted by maximal outdegree are:

%e : o : o o o o o : o o : o :

%e : | : | | / \ / \ / \ : | /|\ : /( )\ :

%e : o : o o o o o o o o : o o o o : o o o o :

%e : | : | / \ | / \ | | : /|\ | : :

%e : o : o o o o o o o o : o o o o : :

%e : | : / \ | | : : :

%e : o : o o o o : : :

%e : | : : : :

%e : o : : : :

%e : : : : :

%e : -1- : ---------------2--------------- : -----3----- : ---4--- :

%e Thus row 5 = [0, 1, 5, 2, 1].

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 1, 2, 1;

%e 0, 1, 5, 2, 1;

%e 0, 1, 10, 6, 2, 1;

%e 0, 1, 22, 16, 6, 2, 1;

%e 0, 1, 45, 43, 17, 6, 2, 1;

%e 0, 1, 97, 113, 49, 17, 6, 2, 1;

%e 0, 1, 206, 300, 136, 50, 17, 6, 2, 1;

%e 0, 1, 450, 787, 386, 142, 50, 17, 6, 2, 1;

%e 0, 1, 982, 2074, 1081, 409, 143, 50, 17, 6, 2, 1;

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

%p `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*

%p b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))

%p end:

%p T:= (n, k)-> b(n-1$2, k$2) -`if`(k=0, 0, b(n-1$2, k-1$2)):

%p seq(seq(T(n, k), k=0..n-1), n=1..14);

%t b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[k == 0, 0, b[n-1, n-1, k-1, k-1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Jul 01 2014, translated from Maple *)

%Y Columns k=2-10 give: A244398, A244399, A244400, A244401, A244402, A244403, A244404, A244405, A244406.

%Y T(2n,n) gives A244407(n).

%Y T(2n+1,n) gives A244410(n).

%Y Row sum give A000081.

%Y Cf. A244454.

%K nonn,tabl

%O 1,9

%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 26 2014