OFFSET
1,10
COMMENTS
An orderless same-tree of weight n > 0 is either a single node of weight n, or a finite multiset of two or more orderless same-trees whose weights are all the same and sum to n.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
EXAMPLE
Triangle begins:
1
1 1
1 0 1
1 1 1 2
1 0 0 0 1
1 1 1 2 1 3
1 0 0 0 0 0 1
1 1 1 3 4 4 3 5
1 0 1 0 1 0 1 0 2
1 1 0 0 1 2 1 1 1 3
1 0 0 0 0 0 0 0 0 0 1
1 1 2 4 5 10 11 14 12 14 7 13
1 0 0 0 0 0 0 0 0 0 0 0 1
1 1 0 0 0 0 1 2 1 1 1 1 1 3
The T(8,5) = 4 orderless same-trees: (4((11)(11))), (4(1111)), ((22)(2(11))), (222(11)).
MATHEMATICA
olstrees[n_]:=Prepend[Join@@Table[Select[Tuples[olstrees/@ptn], OrderedQ], {ptn, Select[IntegerPartitions[n], Length[#]>1&&SameQ@@#&]}], n];
Table[Length[Select[olstrees[n], Count[#, _Integer, {-1}]===k&]], {n, 14}, {k, n}]
PROG
(PARI)
S(g, k)={polcoef(exp(sum(i=1, k, x^i*subst(g, y, y^i)/i) + O(x*x^k)), k)}
A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sumdiv(n, d, S(v[n/d], d))); apply(p -> Vecrev(p/y), v)}
{ my(v=A(16)); for(n=1, #v, print(v[n])) } \\ Andrew Howroyd, Aug 20 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Mar 19 2018
STATUS
approved