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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.
1
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
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
Sequence in context: A032048 A287128 A286062 * A238426 A032066 A107316
KEYWORD
nonn,eigen
AUTHOR
Geoffrey Critzer, Nov 15 2012
STATUS
approved