A320175 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

1, 2, 5, 13, 37, 120, 395, 1381, 4931, 18074, 67287, 254387, 972559, 3756315, 14629237, 57395490, 226613217, 899773355, 3590349661, 14390323014, 57907783039, 233867667197, 947601928915, 3851054528838, 15693587686823, 64114744713845, 262543966114921, 1077406218930902

Offset: 1

Author

Gus Wiseman, Oct 07 2018

Keywords

• nonn

Comments

A rooted tree is series-reduced if every non-leaf node has at least two branches.

Examples

			The a(1) = 1 through a(4) = 13 trees:
  (1)  (2)       (3)            (4)
       ((1)(1))  (21)           (31)
                 ((1)(2))       ((1)(3))
                 ((1)(1)(1))    ((2)(2))
                 ((1)((1)(1)))  ((1)(21))
                                ((1)(1)(2))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)(1)(1)(1))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A317099, A319312, A320173, A320174.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
sot[m_]:=If[UnsameQ@@m,Prepend[#,m],#]&[Join@@Table[Union[Sort/@Tuples[sot/@p]],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[sot[m],{m,IntegerPartitions[n]}]],{n,10}]

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=prod(k=1, n, 1 + x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018