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%I #4 May 24 2013 21:38:41
%S 1,1,1,1,1,2,1,1,2,1,4,1,1,2,2,4,4,6,1,1,2,2,4,1,8,7,2,11,9,1,1,2,2,4,
%T 2,8,7,6,5,21,11,9,24,12,1
%N Table where T(n,k) is the number of unordered trees with n edges (n+1 nodes) whose node out-degrees form the k-th partition of the integer n (in Mathematica order).
%e For the partition [2,1^2] (a(10)=T(4,4)) there are the four trees:
%e ..o.....o.....o.....o
%e ./.\.../.\....|.....|
%e o...o.o...o...o.....o
%e |...|.|....../.\....|
%e o...o.o.....o...o...o
%e ......|.....|....../.\
%e ......o.....o.....o...o
%e Table T(n,k) begins:
%e 1;
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 2, 1, 4, 1;
%e 1, 2, 2, 4, 4, 6, 1;
%e 1, 2, 2, 4, 1, 8, 7, 2, 11, 9, 1;
%e 1, 2, 2, 4, 2, 8, 7, 6, 5, 21, 11, 9, 24, 12, 1;
%Y Cf. A000041 (row lengths), A000081 (row sums), A125181.
%K more,nonn,tabf
%O 0,6
%A _Franklin T. Adams-Watters_, May 15 2008
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