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%I #18 Jan 18 2017 08:58:56
%S 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 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 578,484,94,4,0,1,1020,922,202,11,0,1,1820,1775,412,28
%N 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 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 Alois P. Heinz, <a href="/A245120/b245120.txt">Rows n = 1..140, flattened</a>
%e The A124346(7) = 6 7-node rooted identity trees with thinning limbs sorted by root outdegree are:
%e : o : o o o o : o :
%e : | : / \ / \ / \ / \ : /|\ :
%e : 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 :
%e : | : | | | | : | :
%e : o : o o o o : o :
%e : | : | | | : :
%e : o : o o o : :
%e : | : | : :
%e : o : o : :
%e : | : : :
%e : o : : :
%e : : : :
%e : -1- : -------------2------------ : --3-- :
%e Thus row 7 = [0, 1, 4, 1].
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 1;
%e 0, 1, 3;
%e 0, 1, 4, 1;
%e 0, 1, 8, 2;
%e 0, 1, 12, 4;
%e 0, 1, 22, 9;
%e 0, 1, 36, 17, 2;
%e 0, 1, 63, 35, 3;
%p b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
%p `if`(i<1 or v<1 or n<v, 0, add(binomial(A(i, min(i-1, h)), j)
%p *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v))))
%p end:
%p A:= proc(n, k) option remember;
%p `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k, n-1)))
%p end:
%p g:= proc(n) local k; if n=1 then 0 else
%p for k while T(n, k)>0 do od; k-1 fi
%p end:
%p T:= (n, k)-> b(n-1$2, k$2):
%p seq(seq(T(n, k), k=0..g(n)), n=1..25);
%t 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 Column k=0-10 give: A000007(n-1), A000012 (for n>1), A245121, A245122, A245123, A245124, A245125, A245126, A245127, A245128, A245129.
%Y Row sums give A124346.
%Y Cf. A244657.
%K nonn,tabf
%O 1,14
%A _Alois P. Heinz_, Jul 12 2014
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