A219226 Number of rooted unlabeled ordered (plane) trees with 2n leaves such that i) every internal node has an even number of children and ii) every path from the root to a leaf is the same length.

0, 1, 2, 3, 6, 13, 29, 65, 147, 337, 785, 1857, 4452, 10789, 26365, 64833, 160167, 397025, 986593, 2456193, 6123726, 15286021, 38198573, 95555937, 239294222, 599914489, 1505750425, 3783967201, 9521244242, 23988787485, 60520345765, 152889244033, 386752047956

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2401

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 91

Formula

O.g.f. satisfies A(x) = x + A(x^2/(1-x^2)).

Maple

a:= proc(n) option remember; add(`if`(k=0, 1,
     `if`(k::odd, a((k+1)/2)*binomial(n-1, k), 0)), k=0..n-1)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 26 2022

Mathematica

nn=60; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0 == Series[f[x]-x-f[x^2/(1-x^2)], {x, 0, nn}], x]; a[0]=0; Table[a[n], {n, 0, nn, 2}]/.sol

Crossrefs

Cf. A014535, A027826.

Sequence in context: A032048 A287128 A286062 * A238426 A032066 A107316

Adjacent sequences: A219223 A219224 A219225 * A219227 A219228 A219229

Keyword

nonn,eigen

Author

Geoffrey Critzer, Nov 15 2012

Status

approved