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A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

%I A320172 #8 Oct 26 2018 00:52:35
%S A320172 1,2,5,9,19,38,79,163,352,750,1633,3558,7783,17020,37338,81920,180399,
%T A320172 398600,885101,1975638,4435741,10013855,22726109,51807432,118545425,
%U A320172 272024659,625488420,1440067761,3317675261,7644488052,17610215982,40547552277,93298838972,214516498359,492844378878
%N A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
%C A320172 A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root. In an identity tree, all branches directly under any given node are different.
%H A320172 Andrew Howroyd, <a href="/A320172/b320172.txt">Table of n, a(n) for n = 1..500</a>
%e A320172 The a(1) = 1 through a(5) = 19 rooted identity trees:
%e A320172   (1)  (2)   (3)        (4)         (5)
%e A320172        (11)  (21)       (22)        (32)
%e A320172              (111)      (31)        (41)
%e A320172              ((1)(2))   (211)       (221)
%e A320172              ((1)(11))  (1111)      (311)
%e A320172                         ((1)(3))    (2111)
%e A320172                         ((1)(21))   (11111)
%e A320172                         ((2)(11))   ((1)(4))
%e A320172                         ((1)(111))  ((2)(3))
%e A320172                                     ((1)(31))
%e A320172                                     ((1)(22))
%e A320172                                     ((2)(21))
%e A320172                                     ((3)(11))
%e A320172                                     ((1)(211))
%e A320172                                     ((11)(21))
%e A320172                                     ((2)(111))
%e A320172                                     ((1)(1111))
%e A320172                                     ((11)(111))
%e A320172                                     ((1)(2)(11))
%t A320172 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320172 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320172 gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn],UnsameQ@@#&]],{mtn,Select[mps[m],Length[#]>1&]}],m];
%t A320172 Table[Sum[Length[Select[gig[y],SameQ@@Length/@Position[#,_Integer]&]],{y,Sort /@IntegerPartitions[n]}],{n,8}]
%o A320172 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A320172 seq(n)={my(u=vector(n, n, numbpart(n)), v=vector(n)); while(u, v+=u; u=WeighT(u)-u); v} \\ _Andrew Howroyd_, Oct 25 2018
%Y A320172 Cf. A000669, A005804, A048816, A079500, A119262, A120803, A141268, A292504, A300660, A319312.
%Y A320172 Cf. A320154, A320155, A320160, A320171, A320173, A320177, A320178, A320179.
%K A320172 nonn
%O A320172 1,2
%A A320172 _Gus Wiseman_, Oct 07 2018
%E A320172 Terms a(13) and beyond from _Andrew Howroyd_, Oct 25 2018