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A245120 Number T(n,k) of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=max-index-of-row(n), read by rows.

%I A245120 #18 Jan 18 2017 08:58:56
%S A245120 1,0,1,0,1,0,1,1,0,1,1,0,1,3,0,1,4,1,0,1,8,2,0,1,12,4,0,1,22,9,0,1,36,
%T A245120 17,2,0,1,63,35,3,0,1,107,67,9,0,1,188,131,20,0,1,327,249,46,1,0,1,
%U A245120 578,484,94,4,0,1,1020,922,202,11,0,1,1820,1775,412,28
%N A245120 Number T(n,k) of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=max-index-of-row(n), read by rows.
%C A245120 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 A245120 Alois P. Heinz, <a href="/A245120/b245120.txt">Rows n = 1..140, flattened</a>
%e A245120 The A124346(7) = 6 7-node rooted identity trees with thinning limbs sorted by root outdegree are:
%e A245120 :  o  :   o     o       o      o   :   o   :
%e A245120 :  |  :  / \   / \     / \    / \  :  /|\  :
%e A245120 :  o  : o   o o   o   o   o  o   o : o o o :
%e A245120 :  |  : |     |   |  / \    ( )  | : | |   :
%e A245120 :  o  : o     o   o o   o   o o  o : o o   :
%e A245120 :  |  : |     |     |       |      : |     :
%e A245120 :  o  : o     o     o       o      : o     :
%e A245120 :  |  : |     |     |              :       :
%e A245120 :  o  : o     o     o              :       :
%e A245120 :  |  : |                          :       :
%e A245120 :  o  : o                          :       :
%e A245120 :  |  :                            :       :
%e A245120 :  o  :                            :       :
%e A245120 :     :                            :       :
%e A245120 : -1- : -------------2------------ : --3-- :
%e A245120 Thus row 7 = [0, 1, 4, 1].
%e A245120 Triangle T(n,k) begins:
%e A245120 1;
%e A245120 0, 1;
%e A245120 0, 1;
%e A245120 0, 1,  1;
%e A245120 0, 1,  1;
%e A245120 0, 1,  3;
%e A245120 0, 1,  4,  1;
%e A245120 0, 1,  8,  2;
%e A245120 0, 1, 12,  4;
%e A245120 0, 1, 22,  9;
%e A245120 0, 1, 36, 17, 2;
%e A245120 0, 1, 63, 35, 3;
%p A245120 b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
%p A245120       `if`(i<1 or v<1 or n<v, 0, add(binomial(A(i, min(i-1, h)), j)
%p A245120        *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v))))
%p A245120     end:
%p A245120 A:= proc(n, k) option remember;
%p A245120       `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k, n-1)))
%p A245120     end:
%p A245120 g:= proc(n) local k; if n=1 then 0 else
%p A245120        for k while T(n, k)>0 do od; k-1 fi
%p A245120     end:
%p A245120 T:= (n, k)-> b(n-1$2, k$2):
%p A245120 seq(seq(T(n, k), k=0..g(n)), n=1..25);
%t A245120 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, Sum[Binomial[A[i, Min[i-1, h]], 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]}]]; g[n_] := If[n==1, 0, For[k=1, T[n, k]>0, k++]; k-1]; T[n_, k_] := b[n-1, n-1, k, k]; Table[T[n, k], {n, 1, 25}, {k, 0, g[n]}] // Flatten (* _Jean-François Alcover_, Jan 18 2017, translated from Maple *)
%Y A245120 Column k=0-10 give: A000007(n-1), A000012 (for n>1), A245121, A245122, A245123, A245124, A245125, A245126, A245127, A245128, A245129.
%Y A245120 Row sums give A124346.
%Y A245120 Cf. A244657.
%K A245120 nonn,tabf
%O A245120 1,14
%A A245120 _Alois P. Heinz_, Jul 12 2014