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A217661 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-x)^k.

%I A217661 #20 May 12 2020 18:18:39
%S A217661 1,1,2,6,16,42,114,314,870,2426,6804,19168,54198,153730,437232,
%T A217661 1246480,3560838,10190810,29212432,83860176,241051796,693709896,
%U A217661 1998535892,5763312876,16635018146,48054500898,138923916700,401908892716,1163493516356,3370283517032
%N A217661 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-x)^k.
%C A217661 Radius of convergence of g.f. A(x) is |x| < 0.339332122592...
%C A217661 More generally, given
%C A217661 A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-t*x)^k then
%C A217661 A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).
%H A217661 Vincenzo Librandi, <a href="/A217661/b217661.txt">Table of n, a(n) for n = 0..400</a>
%H A217661 Colin Defant, James Propp, <a href="https://arxiv.org/abs/2002.07144">Quantifying Noninvertibility in Discrete Dynamical Systems</a>, arXiv:2002.07144 [math.CO], 2020.
%F A217661 G.f.: (1-x) / sqrt(1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4).
%F A217661 a(n) ~ (1-r) / (2 * sqrt(Pi*n) * sqrt(1 - 3*r + 2*r^2 - r^3) * r^n), where r = 0.33933212259239... is the root of the equation 1-4*r+4*r^2-4*r^3+4*r^4 = 0. - _Vaclav Kotesovec_, Feb 17 2014
%F A217661 a(n) = Sum_{m=0..n} Sum_{k=0,n-m} C(m,k)^2*C(n-m-1,n-m-k). - _Vladimir Kruchinin_, Jan 16 2018
%e A217661 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 42*x^5 + 114*x^6 + 314*x^7 + ...
%e A217661 where the g.f. equals the series:
%e A217661 A(x) = 1 +
%e A217661 x*(1 + x/(1-x)) +
%e A217661 x^2*(1 + 2^2*x/(1-x) + x^2/(1-x)^2) +
%e A217661 x^3*(1 + 3^2*x/(1-x) + 3^2*x^2/(1-x)^2 + x^3/(1-x)^3) +
%e A217661 x^4*(1 + 4^2*x/(1-x) + 6^2*x^2/(1-x)^2 + 4^2*x^3/(1-x)^3 + x^4/(1-x)^4) +
%e A217661 x^5*(1 + 5^2*x/(1-x) + 10^2*x^2/(1-x)^2 + 10^2*x^3/(1-x)^3 + 5^2*x^4/(1-x)^4 + x^5/(1-x)^5) + ...
%t A217661 CoefficientList[Series[(1-x)/Sqrt[1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 17 2014 *)
%o A217661 (PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k/(1-x +x*O(x^n))^k )), n)}
%o A217661 for(n=0,40,print1(a(n),", "))
%o A217661 (Maxima)
%o A217661 a(n):=sum(sum(binomial(m,k)^2*binomial(n-m-1,n-m-k),k,0,n-m),m,0,n);
%Y A217661 Cf. A217664, A217665, A217666, A217615.
%K A217661 nonn
%O A217661 0,3
%A A217661 _Paul D. Hanna_, Oct 10 2012