A298533 Number of unlabeled rooted trees with n vertices such that every branch of the root has the same number of leaves.
1, 1, 2, 4, 8, 15, 31, 64, 144, 333, 808, 2004, 5109, 13199, 34601, 91539, 244307, 656346, 1774212, 4820356, 13157591, 36060811, 99198470, 273790194, 757971757, 2104222594, 5856496542, 16338140048, 45678276507, 127964625782, 359155302204, 1009790944307
Offset: 1
Keywords
Examples
The a(5) = 8 trees: ((((o)))), (((oo))), ((o(o))), ((ooo)), (o((o))), ((o)(o)), (oo(o)), (oooo)
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..500
Crossrefs
Programs
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Mathematica
rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]]; Table[Length[Select[rut[n],SameQ@@(Count[#,{},{0,Infinity}]&/@#)&]],{n,15}] -
PARI
\\ here R is A055277 as vector of polynomials EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)} R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1 + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)}; seq(n)={my(M=Mat(apply(p->Colrev(p,n), R(n-1)))); concat([1],sum(i=2, #M, EulerT(M[i,])))} \\ Andrew Howroyd, May 20 2018
Extensions
Terms a(19) and beyond from Andrew Howroyd, May 20 2018