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%I #28 May 20 2021 22:40:01
%S 0,1,1,3,8,25,80,267,911,3170,11192,39993,144320,525124,1924196,
%T 7093603,26288928,97878831,365918064,1372982706,5168555770,
%U 19514482964,73876936272,280363191353,1066357904128,4064204607372
%N G.f. satisfies A(x) = x*(1+A^2)^2/(1-A+A^2).
%H Alois P. Heinz, <a href="/A101490/b101490.txt">Table of n, a(n) for n = 0..600</a>
%H M. Bousquet-Mélou, <a href="http://arXiv.org/abs/math.CO/0501266">Limit laws for embedded trees</a>
%H Cyril Banderier, Markus Kuba, and Michael Wallner, <a href="https://arxiv.org/abs/2103.03751">Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions</a>, arXiv:2103.03751 [math.PR], 2021.
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 413
%F G.f: x*c(x)*c(x^2*c(x)^2), c(x) the g.f. of A000108. - _Paul Barry_, Jun 02 2009]
%F a(n+1) = Sum_{k, 0<=k<=[n/2]} A039599(n-k,k)*A000108(k). - _Philippe Deléham_, Apr 12 2007
%F a(n) ~ 2^(2*n-3/2)/(Gamma(3/4)*n^(5/4)) * (1 - Gamma(3/4)/ (n^(1/4)*sqrt(Pi/2)) + 9*Gamma(3/4)^2/(4*sqrt(2*n)*Pi)). - _Vaclav Kotesovec_, Sep 16 2013
%F a(n) = T(2*n-1,n)/n, where T(n,k)=T(n-1,k)-T(n-2,k)+T(n-1,k-1)+2*T(n-3,k-1)+T(n-5,k-1). - _Vladimir Kruchinin_, Sep 29 2014
%p a:= n-> coeff(series(RootOf(A=x*(1+A^2)^2/(1-A+A^2)
%p , A), x, n+1), x, n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 16 2013
%t For[A = 1; n = 1, n <= 26, n++, A = x*(1+A^2)^2/(1-A+A^2) + O[x]^n]; CoefficientList[A, x] (* _Jean-François Alcover_, Jun 29 2011, updated Apr 23 2016 *)
%o (Maxima)
%o T(n,k):= if n<0 then 0 else if n=k then 1 else if n>0 and k=0 then 0 else T(n-1,k)-T(n-2,k)+T(n-1,k-1)+2*T(n-3,k-1)+T(n-5,k-1);
%o makelist(T(2*n-1,n)/n,n,1,7); /* _Vladimir Kruchinin_, Sep 29 2014 */
%K nonn
%O 0,4
%A _Ralf Stephan_, Jan 21 2005
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