OFFSET
1,8
COMMENTS
An r-tree (A093637) of size n > 0 is a finite sequence of r-trees with weakly decreasing sizes summing to n - 1. This is a similar construction to p-trees (A196545) except that r-trees are not required to be series-reduced and are weighted by all nodes (including the root) rather than just the leaves.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
EXAMPLE
Triangle begins:
1
1 0
1 1 0
1 2 1 0
1 4 3 1 0
1 6 8 4 1 0
1 9 19 14 5 1 0
1 12 36 40 21 6 1 0
1 16 65 102 75 30 7 1 0
1 20 106 223 224 123 40 8 1 0
1 25 168 457 604 439 191 52 9 1 0
...
The T(6,3) = 8 r-trees: (((ooo))), (((oo)o)), (((o)oo)), (((oo))o), (((o)o)o), ((oo)(o)), (((o))oo), ((o)(o)o).
MATHEMATICA
rtrees[n_]:=Join@@Table[Tuples[rtrees/@y], {y, IntegerPartitions[n-1]}];
Table[Length[Select[rtrees[n], Count[#, {}, {-2}]===k&]], {n, 8}, {k, n}]
PROG
(PARI) A(n)={my(v=vector(n)); v[1]=y; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); vector(n, k, Vecrev(v[k]/y, k))}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Mar 20 2018
STATUS
approved