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A316977 Number of series-reduced rooted trees whose leaves are {1, 1, 2, 2, 3, 3, ..., n, n}.

%I A316977 #11 Jan 02 2021 22:44:39
%S A316977 1,12,575,66080,13830706,4566898564,2181901435364,1422774451251512,
%T A316977 1213875872220833664,1312273759143855989808,1752860078230602866012288,
%U A316977 2834766624822130489716563008,5458358420687156358967526721408,12339106957086349462329140342122112
%N A316977 Number of series-reduced rooted trees whose leaves are {1, 1, 2, 2, 3, 3, ..., n, n}.
%C A316977 A rooted tree is series-reduced if every non-leaf node has at least two branches.
%F A316977 a(n) = A292505(A061742(n)). - _Andrew Howroyd_, Nov 19 2018
%e A316977 The a(2) = 12 trees are (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122).
%t A316977 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A316977 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A316977 gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
%t A316977 Table[Length[gro[Ceiling[Range[1/2,n,1/2]]]],{n,4}]
%o A316977 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A316977 cycleIndexSeries(n)={my(v=vector(2*n), vars=vector(2*n-2,i,sv(2+i))); v[1]=sv(1); for(n=2, #v, v[n] = substvec(polcoef( sExp(x*Ser(v[1..n])), n ), vars[1..n-2], vector(n-2))); sCartProd(x*Ser(v), 1/(1-x^2*symGroupCycleIndex(2)) + O(x*x^(2*n)))}
%o A316977 seq(n)={my(p=substvec(cycleIndexSeries(n), [sv(1), sv(2)], [1,1])); vector(n, n, polcoef(p,2*n))} \\ _Andrew Howroyd_, Jan 02 2021
%Y A316977 Cf. A000081, A000669, A007717, A020555, A050535, A094574, A292504, A316655, A316972, A316974.
%K A316977 nonn
%O A316977 1,2
%A A316977 _Gus Wiseman_, Jul 17 2018
%E A316977 Terms a(6) and beyond from _Andrew Howroyd_, Jan 02 2021