A320175 - OEIS

%I #10 Oct 25 2018 22:19:35

%S 1,2,5,13,37,120,395,1381,4931,18074,67287,254387,972559,3756315,

%T 14629237,57395490,226613217,899773355,3590349661,14390323014,

%U 57907783039,233867667197,947601928915,3851054528838,15693587686823,64114744713845,262543966114921,1077406218930902

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

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

%H Andrew Howroyd, <a href="/A320175/b320175.txt">Table of n, a(n) for n = 1..200</a>

%e The a(1) = 1 through a(4) = 13 trees:

%e   (1)  (2)       (3)            (4)

%e        ((1)(1))  (21)           (31)

%e                  ((1)(2))       ((1)(3))

%e                  ((1)(1)(1))    ((2)(2))

%e                  ((1)((1)(1)))  ((1)(21))

%e                                 ((1)(1)(2))

%e                                 ((1)((1)(2)))

%e                                 ((2)((1)(1)))

%e                                 ((1)(1)(1)(1))

%e                                 ((1)((1)(1)(1)))

%e                                 ((1)(1)((1)(1)))

%e                                 (((1)(1))((1)(1)))

%e                                 ((1)((1)((1)(1))))

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t sot[m_]:=If[UnsameQ@@m,Prepend[#,m],#]&[Join@@Table[Union[Sort/@Tuples[sot/@p]],{p,Select[mps[m],Length[#]>1&]}]];

%t Table[Length[Join@@Table[sot[m],{m,IntegerPartitions[n]}]],{n,10}]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o 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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 07 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Oct 25 2018