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A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

%I A244925 #16 Jun 01 2021 15:43:07
%S A244925 1,0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,2,2,1,1,0,1,4,3,2,1,1,0,1,4,5,3,2,
%T A244925 1,1,0,1,7,7,6,3,2,1,1,0,1,8,12,8,6,3,2,1,1,0,1,12,18,15,9,6,3,2,1,1,
%U A244925 0,1,14,27,23,16,9,6,3,2,1,1,0,1,21,42,39,26,17,9,6,3,2,1,1
%N A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
%H A244925 Alois P. Heinz, <a href="/A244925/b244925.txt">Rows n = 1..141, flattened</a>
%e A244925 The A048816(5) = 5 rooted trees with 5 nodes with every leaf at the same height sorted by height are:
%e A244925   :    o    :   o     o   :   o   :  o  :
%e A244925   :  /( )\  :  / \    |   :   |   :  |  :
%e A244925   : o o o o : o   o   o   :   o   :  o  :
%e A244925   :         : |   |  /|\  :   |   :  |  :
%e A244925   :         : o   o o o o :   o   :  o  :
%e A244925   :         :             :  / \  :  |  :
%e A244925   :         :             : o   o :  o  :
%e A244925   :         :             :       :  |  :
%e A244925   :         :             :       :  o  :
%e A244925   :         :             :       :     :
%e A244925   : ---1--- : -----2----- : --3-- : -4- :
%e A244925 Thus row 5 = [0, 1, 2, 1, 1].
%e A244925 Triangle T(n,k) begins:
%e A244925   1;
%e A244925   0, 1;
%e A244925   0, 1,  1;
%e A244925   0, 1,  1,  1;
%e A244925   0, 1,  2,  1,  1;
%e A244925   0, 1,  2,  2,  1,  1;
%e A244925   0, 1,  4,  3,  2,  1, 1;
%e A244925   0, 1,  4,  5,  3,  2, 1, 1;
%e A244925   0, 1,  7,  7,  6,  3, 2, 1, 1;
%e A244925   0, 1,  8, 12,  8,  6, 3, 2, 1, 1;
%e A244925   0, 1, 12, 18, 15,  9, 6, 3, 2, 1, 1;
%e A244925   0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1;
%e A244925   ...
%p A244925 with(numtheory):
%p A244925 T:= proc(n, k) option remember; `if`(n=1, 1, `if`(k=0, 0,
%p A244925       add(add(`if`(d<k, 0, T(d, k-1)*d), d=divisors(j))*
%p A244925       T(n-j, k), j=1..n-1)/(n-1)))
%p A244925     end:
%p A244925 seq(seq(T(n, k), k=0..n-1), n=1..14);
%t A244925 T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[ Sum[ If[d<k, 0, T[d, k-1]*d], {d, Divisors[j]}] * T[n-j, k], {j, 1, n-1}]/(n-1)]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Jan 28 2015, after _Alois P. Heinz_ *)
%Y A244925 Columns k=0-10 give: A000007(n-1), A000012 (for n>0), A002865(n-1) (for n>2), A048808, A048809, A048810, A048811, A048812, A048813, A048814, A048815.
%Y A244925 T(2n+1,n) gives A074045.
%Y A244925 Row sums give A048816.
%K A244925 nonn,tabl
%O A244925 1,13
%A A244925 _Alois P. Heinz_, Jul 08 2014