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A217282 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^k ). 1

%I

%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 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).

%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 +...

%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

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)