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A320176 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is a strict integer partition of n.

%I A320176 #8 Oct 26 2018 17:09:51
%S A320176 1,1,3,3,5,13,15,23,33,99,109,183,251,383,1071,1261,2007,2875,4291,
%T A320176 5829,16297,18563,30313,42243,63707,85351,125465,297843,356657,556729,
%U A320176 783637,1151803,1564173,2249885,2988729,6803577,8026109,12465665,17124495,25272841,33657209
%N A320176 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is a strict integer partition of n.
%C A320176 Also the number of orderless tree-factorizations of Heinz numbers of strict integer partitions of n.
%C A320176 Also the number of phylogenetic trees on a set of distinct labels summing to n.
%H A320176 Andrew Howroyd, <a href="/A320176/b320176.txt">Table of n, a(n) for n = 1..1000</a>
%F A320176 a(n) = Sum_{k>0} A008289(n, k)*A005804(k). - _Andrew Howroyd_, Oct 26 2018
%e A320176 The a(1) = 1 through a(7) = 15 rooted trees:
%e A320176   (1)  (2)  (3)       (4)       (5)       (6)            (7)
%e A320176             (21)      (31)      (32)      (42)           (43)
%e A320176             ((1)(2))  ((1)(3))  (41)      (51)           (52)
%e A320176                                 ((1)(4))  (321)          (61)
%e A320176                                 ((2)(3))  ((1)(5))       (421)
%e A320176                                           ((2)(4))       ((1)(6))
%e A320176                                           ((1)(23))      ((2)(5))
%e A320176                                           ((2)(13))      ((3)(4))
%e A320176                                           ((3)(12))      ((1)(24))
%e A320176                                           ((1)(2)(3))    ((2)(14))
%e A320176                                           ((1)((2)(3)))  ((4)(12))
%e A320176                                           ((2)((1)(3)))  ((1)(2)(4))
%e A320176                                           ((3)((1)(2)))  ((1)((2)(4)))
%e A320176                                                          ((2)((1)(4)))
%e A320176                                                          ((4)((1)(2)))
%t A320176 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320176 got[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[got/@p]],{p,Select[sps[m],Length[#]>1&]}],m];
%t A320176 Table[Length[Join@@Table[got[m],{m,Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,20}]
%o A320176 (PARI) \\ here S(n) is first n terms of A005804.
%o A320176 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320176 b(n,k)={my(v=vector(n)); for(n=1, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); v}
%o A320176 S(n)={my(M=Mat(vectorv(n, k, b(n,k)))); vector(n, k, sum(i=1, k, binomial(k, i)*(-1)^(k-i)*M[i,k]))}
%o A320176 seq(n)={my(u=S((sqrtint(8*n+1)-1)\2)); [sum(i=1, poldegree(p), polcoef(p,i)*u[i]) | p <- Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))-1)]} \\ _Andrew Howroyd_, Oct 26 2018
%Y A320176 Cf. A000669, A005804, A008289, A141268, A292504, A300660, A319312, A320171, A320174, A320175, A320177, A320178.
%K A320176 nonn
%O A320176 1,3
%A A320176 _Gus Wiseman_, Oct 07 2018
%E A320176 Terms a(31) and beyond from _Andrew Howroyd_, Oct 26 2018