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A244657 Number T(n,k) of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

%I A244657 #35 Sep 07 2018 21:18:53
%S A244657 1,0,1,0,1,1,0,1,1,1,0,1,3,1,1,0,1,4,3,1,1,0,1,9,6,3,1,1,0,1,13,13,6,
%T A244657 3,1,1,0,1,26,25,15,6,3,1,1,0,1,42,55,29,15,6,3,1,1,0,1,81,107,68,31,
%U A244657 15,6,3,1,1,0,1,138,224,140,72,31,15,6,3,1,1
%N A244657 Number T(n,k) of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
%C A244657 In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.
%H A244657 Alois P. Heinz, <a href="/A244657/b244657.txt">Rows n = 1..141, flattened</a>
%e A244657 The A124343(5) = 6 5-node rooted trees with thinning limbs sorted by root outdegree are:
%e A244657 :  o  :   o       o     o   :   o   :    o    :
%e A244657 :  |  :  / \     / \   / \  :  /|\  :  /( )\  :
%e A244657 :  o  : o   o   o   o o   o : o o o : o o o o :
%e A244657 :  |  : |      / \    |   | : |     :         :
%e A244657 :  o  : o     o   o   o   o : o     :         :
%e A244657 :  |  : |                   :       :         :
%e A244657 :  o  : o                   :       :         :
%e A244657 :  |  :                     :       :         :
%e A244657 :  o  :                     :       :         :
%e A244657 :     :                     :       :         :
%e A244657 : -1- : ---------2--------- : --3-- : ---4--- :
%e A244657 Thus row 5 = [0, 1, 3, 1, 1].
%e A244657 Triangle T(n,k) begins:
%e A244657   1;
%e A244657   0, 1;
%e A244657   0, 1,  1;
%e A244657   0, 1,  1,   1;
%e A244657   0, 1,  3,   1,  1;
%e A244657   0, 1,  4,   3,  1,  1;
%e A244657   0, 1,  9,   6,  3,  1,  1;
%e A244657   0, 1, 13,  13,  6,  3,  1, 1;
%e A244657   0, 1, 26,  25, 15,  6,  3, 1, 1;
%e A244657   0, 1, 42,  55, 29, 15,  6, 3, 1, 1;
%e A244657   0, 1, 81, 107, 68, 31, 15, 6, 3, 1, 1;
%p A244657 b:= proc(n, i, h, v) option remember; `if`(n=0,
%p A244657       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A244657       `if`(n=v, 1, add(binomial(A(i, min(i-1, h))+j-1, j)
%p A244657        *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
%p A244657     end:
%p A244657 A:= proc(n, k) option remember;
%p A244657       `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k,n-1)))
%p A244657     end:
%p A244657 T:= (n, k)-> b(n-1$2, k$2):
%p A244657 seq(seq(T(n, k), k=0..n-1), n=1..14);
%t A244657 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, Min[i-1, 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, 1, Min[k, n-1]}]]; T[n_, k_] := b[n-1, n-1, k, k]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Feb 03 2015, after _Alois P. Heinz_ *)
%Y A244657 Columns k=0-10 give: A000007(n-1), A000012 (for n>1), A244703, A244704, A244705, A244706, A244707, A244708, A244709, A244710, A244711.
%Y A244657 T(2n,n) gives A244712.
%Y A244657 Row sums give A124343.
%Y A244657 Cf. A245120, A245151.
%K A244657 nonn,tabl
%O A244657 1,13
%A A244657 _Alois P. Heinz_, Jul 04 2014