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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 A244372 #23 Sep 07 2018 17:22:07
%S A244372 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 A244372 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 A244372 787,386,142,50,17,6,2,1,0,1,982,2074,1081,409,143,50,17,6,2,1
%N 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.
%H A244372 Alois P. Heinz, <a href="/A244372/b244372.txt">Rows n = 1..141, flattened</a>
%e A244372 The A000081(5) = 9 rooted trees with 5 nodes sorted by maximal outdegree are:
%e A244372 :  o  :   o     o     o       o     o   :   o     o   :    o    :
%e A244372 :  |  :   |     |    / \     / \   / \  :   |    /|\  :  /( )\  :
%e A244372 :  o  :   o     o   o   o   o   o o   o :   o   o o o : o o o o :
%e A244372 :  |  :   |    / \  |      / \    |   | :  /|\  |     :         :
%e A244372 :  o  :   o   o   o o     o   o   o   o : o o o o     :         :
%e A244372 :  |  :  / \  |     |                   :             :         :
%e A244372 :  o  : o   o o     o                   :             :         :
%e A244372 :  |  :                                 :             :         :
%e A244372 :  o  :                                 :             :         :
%e A244372 :     :                                 :             :         :
%e A244372 : -1- : ---------------2--------------- : -----3----- : ---4--- :
%e A244372 Thus row 5 = [0, 1, 5, 2, 1].
%e A244372 Triangle T(n,k) begins:
%e A244372   1;
%e A244372   0,  1;
%e A244372   0,  1,   1;
%e A244372   0,  1,   2,    1;
%e A244372   0,  1,   5,    2,    1;
%e A244372   0,  1,  10,    6,    2,   1;
%e A244372   0,  1,  22,   16,    6,   2,   1;
%e A244372   0,  1,  45,   43,   17,   6,   2,  1;
%e A244372   0,  1,  97,  113,   49,  17,   6,  2,  1;
%e A244372   0,  1, 206,  300,  136,  50,  17,  6,  2,  1;
%e A244372   0,  1, 450,  787,  386, 142,  50, 17,  6,  2,  1;
%e A244372   0,  1, 982, 2074, 1081, 409, 143, 50, 17,  6,  2,  1;
%p A244372 b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
%p A244372       `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p A244372        b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
%p A244372     end:
%p A244372 T:= (n, k)-> b(n-1$2, k$2) -`if`(k=0, 0, b(n-1$2, k-1$2)):
%p A244372 seq(seq(T(n, k), k=0..n-1), n=1..14);
%t A244372 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 A244372 Columns k=2-10 give: A244398, A244399, A244400, A244401, A244402, A244403, A244404, A244405, A244406.
%Y A244372 T(2n,n) gives A244407(n).
%Y A244372 T(2n+1,n) gives A244410(n).
%Y A244372 Row sum give A000081.
%Y A244372 Cf. A244454.
%K A244372 nonn,tabl
%O A244372 1,9
%A A244372 _Joerg Arndt_ and _Alois P. Heinz_, Jun 26 2014