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A245151 Number T(n,k) of n-node unlabeled rooted trees with thickening limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

%I A245151 #20 Jan 27 2015 05:02:48
%S A245151 1,0,1,0,1,1,0,2,0,1,0,3,1,0,1,0,5,1,0,0,1,0,7,3,1,0,0,1,0,12,3,1,0,0,
%T A245151 0,1,0,17,8,1,1,0,0,0,1,0,28,9,3,1,0,0,0,0,1,0,42,21,3,1,1,0,0,0,0,1,
%U A245151 0,69,28,5,1,1,0,0,0,0,0,1,0,105,56,9,3,1,1,0,0,0,0,0,1
%N A245151 Number T(n,k) of n-node unlabeled rooted trees with thickening limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
%C A245151 In a rooted tree with thickening limbs the outdegree of a parent node is smaller than or equal to the outdegree of any of its non-leaf child nodes.
%C A245151 T(n+1,1) = Sum_{k=0..n-1} T(n,k) for n>=1.
%C A245151 T(n+1,n) = T(2n+1,n) = 1 for n>=0.
%C A245151 T(n,1+floor((n-1)/2)) = 0 for n>3.
%H A245151 Alois P. Heinz, <a href="/A245151/b245151.txt">Rows n = 1..141, flattened</a>
%e A245151 The A245152(5) = 5 5-node rooted trees with thickening limbs sorted by root outdegree are:
%e A245151 : o   o     o   :     o   :    o    :
%e A245151 : |   |     |   :    / \  :  /( )\  :
%e A245151 : o   o     o   :   o   o : o o o o :
%e A245151 : |   |    /|\  :  / \    :         :
%e A245151 : o   o   o o o : o   o   :         :
%e A245151 : |  / \        :         :         :
%e A245151 : o o   o       :         :         :
%e A245151 : |             :         :         :
%e A245151 : o             :         :         :
%e A245151 :               :         :         :
%e A245151 : ------1------ : ---2--- : ---4--- :
%e A245151 Thus row 5 = [0, 3, 1, 0, 1].
%e A245151 Triangle T(n,k) begins:
%e A245151 1;
%e A245151 0,   1;
%e A245151 0,   1,  1;
%e A245151 0,   2,  0,  1;
%e A245151 0,   3,  1,  0, 1;
%e A245151 0,   5,  1,  0, 0, 1;
%e A245151 0,   7,  3,  1, 0, 0, 1;
%e A245151 0,  12,  3,  1, 0, 0, 0, 1;
%e A245151 0,  17,  8,  1, 1, 0, 0, 0, 1;
%e A245151 0,  28,  9,  3, 1, 0, 0, 0, 0, 1;
%e A245151 0,  42, 21,  3, 1, 1, 0, 0, 0, 0, 1;
%e A245151 0,  69, 28,  5, 1, 1, 0, 0, 0, 0, 0, 1;
%e A245151 0, 105, 56,  9, 3, 1, 1, 0, 0, 0, 0, 0, 1;
%e A245151 0, 176, 81, 12, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;
%p A245151 b:= proc(n, i, h, v) option remember; `if`(n=0,
%p A245151       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A245151       `if`(n=v, 1, add(binomial(A(i, h)+j-1, j)*
%p A245151        b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
%p A245151     end:
%p A245151 A:= proc(n, k) option remember;
%p A245151       `if`(n<2, n, add(b(n-1$2, j$2), j=k..n-1))
%p A245151     end:
%p A245151 T:= (n, k)-> b(n-1$2, k$2):
%p A245151 seq(seq(T(n, k), k=0..n-1), n=1..20);
%t A245151 b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i<1 || v<1 || n<v, 0, If[n == v, 1, Sum[Binomial[A[i, h] + j - 1, j]*b[n - i*j, i-1, h, v-j], {j, 0, Min[n/i, v]}]]]]; A[n_, k_] := A[n, k] = If[n<2, n, Sum[b[n-1, n-1, j, j], {j, k, n-1}]]; T[n_, k_] := b[n-1, n-1, k, k]; Table[ Table[T[n, k], {k, 0, n - 1}], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Jan 27 2015, after _Alois P. Heinz_ *)
%Y A245151 Columns k=0-10 give: A000007(n-1), A245152(n-1), A245142, A245143, A245144, A245145, A245146, A245147, A245148, A245149, A245150.
%Y A245151 Row sums give A245152.
%Y A245151 Cf. A244657 (thinning limbs).
%K A245151 nonn,tabl
%O A245151 1,8
%A A245151 _Alois P. Heinz_, Jul 12 2014