%I #18 Oct 24 2023 12:39:42
%S 1,1,2,3,5,9,16,30,57,110,216,428,857,1730,3516,7191,14785,30544,
%T 63370,131976,275811,578219,1215680,2562652,5415163,11468455,24338744,
%U 51752029,110239033,235218046,502674172,1075823427,2305661425,4947834665,10630848122,22867799427
%N G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^k ).
%C Radius of convergence of g.f. A(x) is r = 0.446171506758870... where 1-r-2*r^2-2*r^3+r^4-r^5 = 0, with A(r) = (1-r^2)/(2*r^3) = 4.5087858050...
%H Michael De Vlieger, <a href="/A217282/b217282.txt">Table of n, a(n) for n = 0..2866</a>
%H Andrei Asinowski, Cyril Banderier, Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2310.12497">Motzkin paths of bounded height with two forbidden contiguous subwords of length two</a>, arXiv:2310.12497 [math.CO], 2023.
%F G.f.: (1-x^2 - sqrt( (1-x-2*x^2-2*x^3+x^4-x^5)/(1-x) ))/(2*x^3).
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 16*x^6 + 30*x^7 +...
%t CoefficientList[Series[(1 - x^2 - Sqrt[(1 - x - 2*x^2 - 2*x^3 + x^4 - x^5)/(1 - x)])/(2*x^3), {x, 0, 35}], x] (* _Michael De Vlieger_, Oct 24 2023 *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(1-x)^k)+x*O(x^n))),n)}
%o (PARI) {a(n)=polcoeff((1-x^2 - sqrt( (1-x-2*x^2-2*x^3+x^4-x^5)/(1-x +x^4*O(x^n)) ))/(2*x^3), n)}
%o for(n=0,40,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 29 2012