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

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Example

The a(5) = 8 trees: ((((o)))), (((oo))), ((o(o))), ((ooo)), (o((o))), ((o)(o)), (oo(o)), (oooo)

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}]

Prog

(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

Crossrefs

Cf. A000081, A003238, A004111, A032305, A289079, A290689, A291443, A297791, A298422, A298534, A298535.

Sequence in context: A300520 A243082 A092603 * A259805 A086125 A227236

Adjacent sequences: A298530 A298531 A298532 * A298534 A298535 A298536

Keyword

nonn

Author

Gus Wiseman, Jan 20 2018

Extensions

Terms a(19) and beyond from Andrew Howroyd, May 20 2018

Status

approved