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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. 12

%I

%S 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 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 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 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 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 T(n+1,1) = Sum_{k=0..n-1} T(n,k) for n>=1.

%C T(n+1,n) = T(2n+1,n) = 1 for n>=0.

%C T(n,1+floor((n-1)/2)) = 0 for n>3.

%H Alois P. Heinz, <a href="/A245151/b245151.txt">Rows n = 1..141, flattened</a>

%e The A245152(5) = 5 5-node rooted trees with thickening limbs sorted by root outdegree are:

%e : o o o : o : o :

%e : | | | : / \ : /( )\ :

%e : o o o : o o : o o o o :

%e : | | /|\ : / \ : :

%e : o o o o o : o o : :

%e : | / \ : : :

%e : o o o : : :

%e : | : : :

%e : o : : :

%e : : : :

%e : ------1------ : ---2--- : ---4--- :

%e Thus row 5 = [0, 3, 1, 0, 1].

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 2, 0, 1;

%e 0, 3, 1, 0, 1;

%e 0, 5, 1, 0, 0, 1;

%e 0, 7, 3, 1, 0, 0, 1;

%e 0, 12, 3, 1, 0, 0, 0, 1;

%e 0, 17, 8, 1, 1, 0, 0, 0, 1;

%e 0, 28, 9, 3, 1, 0, 0, 0, 0, 1;

%e 0, 42, 21, 3, 1, 1, 0, 0, 0, 0, 1;

%e 0, 69, 28, 5, 1, 1, 0, 0, 0, 0, 0, 1;

%e 0, 105, 56, 9, 3, 1, 1, 0, 0, 0, 0, 0, 1;

%e 0, 176, 81, 12, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;

%p b:= proc(n, i, h, v) option remember; `if`(n=0,

%p `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,

%p `if`(n=v, 1, add(binomial(A(i, h)+j-1, 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=k..n-1))

%p end:

%p T:= (n, k)-> b(n-1$2, k$2):

%p seq(seq(T(n, k), k=0..n-1), n=1..20);

%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, 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-Fran├žois Alcover_, Jan 27 2015, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000007(n-1), A245152(n-1), A245142, A245143, A245144, A245145, A245146, A245147, A245148, A245149, A245150.

%Y Row sums give A245152.

%Y Cf. A244657 (thinning limbs).

%K nonn,tabl

%O 1,8

%A _Alois P. Heinz_, Jul 12 2014

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Last modified November 14 07:33 EST 2019. Contains 329111 sequences. (Running on oeis4.)