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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.

%I A219226 #11 Feb 26 2022 13:46:39
%S A219226 0,1,2,3,6,13,29,65,147,337,785,1857,4452,10789,26365,64833,160167,
%T A219226 397025,986593,2456193,6123726,15286021,38198573,95555937,239294222,
%U A219226 599914489,1505750425,3783967201,9521244242,23988787485,60520345765,152889244033,386752047956
%N 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.
%H A219226 Alois P. Heinz, <a href="/A219226/b219226.txt">Table of n, a(n) for n = 0..2401</a>
%H A219226 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 91
%F A219226 O.g.f. satisfies A(x) = x + A(x^2/(1-x^2)).
%p A219226 a:= proc(n) option remember; add(`if`(k=0, 1,
%p A219226      `if`(k::odd, a((k+1)/2)*binomial(n-1, k), 0)), k=0..n-1)
%p A219226     end:
%p A219226 seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 26 2022
%t A219226 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
%Y A219226 Cf. A014535, A027826.
%K A219226 nonn,eigen
%O A219226 0,3
%A A219226 _Geoffrey Critzer_, Nov 15 2012