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%I #18 Sep 07 2018 21:15:38
%S 1,0,1,0,1,1,0,3,0,1,0,7,1,0,1,0,17,2,0,0,1,0,42,4,1,0,0,1,0,105,7,2,
%T 0,0,0,1,0,267,15,2,1,0,0,0,1,0,684,28,4,2,0,0,0,0,1,0,1775,56,7,2,1,
%U 0,0,0,0,1,0,4639,110,12,2,2,0,0,0,0,0,1
%N Number T(n,k) of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
%C T(1,0) = 1 by convention.
%C Sum_{i=2..n-1} T(n,i) = A001678(n+1) for n>1.
%H Alois P. Heinz, <a href="/A244454/b244454.txt">Rows n = 1..141, flattened</a>
%e The A000081(5) = 9 rooted trees with 5 nodes sorted by minimal outdegree of inner nodes are:
%e : o o o o o o o : o : o :
%e : | | | / \ / \ | /|\ : / \ : /( )\ :
%e : o o o o o 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 : o o : :
%e : | / \ | | : : :
%e : o o o o o : : :
%e : | : : :
%e : o : : :
%e : : : :
%e : ------------------1------------------ : ---2--- : ---4--- :
%e Thus row 5 = [0, 7, 1, 0, 1].
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 3, 0, 1;
%e 0, 7, 1, 0, 1;
%e 0, 17, 2, 0, 0, 1;
%e 0, 42, 4, 1, 0, 0, 1;
%e 0, 105, 7, 2, 0, 0, 0, 1;
%e 0, 267, 15, 2, 1, 0, 0, 0, 1;
%e 0, 684, 28, 4, 2, 0, 0, 0, 0, 1;
%e 0, 1775, 56, 7, 2, 1, 0, 0, 0, 0, 1;
%e 0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1;
%p b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
%p 1, 0), `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p b(n-i*j, i-1, max(0, t-j), k), j=0..n/i)))
%p end:
%p T:= (n, k)-> b(n-1$2, k$2) -`if`(n=1 and k=0, 0, b(n-1$2, k+1$2)):
%p seq(seq(T(n, k), k=0..n-1), n=1..14);
%t b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i<1, 0, Sum[Binomial[b[i-1, i-1, k, k]+j-1, j]* b[n-i*j, i-1, Max[0, t-j], k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Jan 08 2015, translated from Maple *)
%Y Columns k=0-10 give: A063524, A244455, A244456, A244457, A244458, A244459, A244460, A244461, A244462, A244463, A244464.
%Y Row sums give A000081.
%Y Cf. A001678, A244372, A244530 (ordered unlabeled rooted trees).
%K nonn,tabl
%O 1,8
%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 28 2014
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