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

Author

Geoffrey Critzer, Nov 15 2012

Keywords

• nonn
• eigen

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2401
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 91

Crossrefs

Cf. A014535, A027826.

Programs

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

Formula

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