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

1, 12, 575, 66080, 13830706, 4566898564, 2181901435364, 1422774451251512, 1213875872220833664, 1312273759143855989808, 1752860078230602866012288, 2834766624822130489716563008, 5458358420687156358967526721408, 12339106957086349462329140342122112

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..14.

Formula

a(n) = A292505(A061742(n)). - Andrew Howroyd, Nov 19 2018

Example

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).

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]]]];
gro[m_]:=If[Length[m]==1, m, Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])]];
Table[Length[gro[Ceiling[Range[1/2, n, 1/2]]]], {n, 4}]

Prog

(PARI) \\ See links in A339645 for combinatorial species functions.
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)))}
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

Crossrefs

Cf. A000081, A000669, A007717, A020555, A050535, A094574, A292504, A316655, A316972, A316974.

Sequence in context: A192605 A192606 A179424 * A283490 A159722 A307948

Adjacent sequences: A316974 A316975 A316976 * A316978 A316979 A316980

Keyword

nonn

Author

Gus Wiseman, Jul 17 2018

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jan 02 2021

Status

approved