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%I #16 Oct 30 2022 08:59:57
%S 1,2,7,27,114,509,2365,11318,55411,276231,1397430,7156089,37023225,
%T 193229466,1016141199,5378940051,28638955098,153267403397,
%U 824014568581,4448456379134,24104579252971,131055735586767,714741620026542,3908997981612017,21434123083817329
%N Expansion of o.g.f. x*s(x)/(1-x*s(x)-x^2*s(x)^2), where s(x) is the o.g.f. of the little Schroeder numbers (A001003).
%H Andrew Howroyd, <a href="/A180473/b180473.txt">Table of n, a(n) for n = 1..500</a>
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F a(n) = Sum_{k=1..n} (k/(2^k*n))*(Sum_{j=0..n-k} binomial(n,j)*2^(n-j)*(-1)^j*binomial(2*n-k-j-1, n-1))*Fibonacci(k).
%o (Maxima) a(n):=sum(k/(2^k*n)*sum(binomial(n,j)*2^(n-j)*(-1)^j*binomial(2*n-k-j-1,n-1),j,0,n-k)*fib(k),k,1,n);
%o (PARI) seq(n)={my(p=x*(1+x-sqrt(1 - 6*x + x^2 + O(x*x^n)))/(4*x)); Vec(p/(1 - p - p^2))} \\ _Andrew Howroyd_, Apr 17 2021
%K nonn
%O 1,2
%A _Vladimir Kruchinin_, Sep 07 2010
%E Terms a(21) and beyond from _Andrew Howroyd_, Apr 17 2021