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A184018 Expansion of c(x/(1-x-x^2)) / (1-x-x^2), c(x) the g.f. of A000108.

%I A184018 #31 May 23 2025 07:51:34
%S A184018 1,2,6,19,67,254,1017,4236,18168,79680,355635,1609912,7373401,
%T A184018 34102976,159055728,747211753,3532452169,16792693562,80224098381,
%U A184018 384948157635,1854469572120,8965866981294,43488834409737,211569299607282
%N A184018 Expansion of c(x/(1-x-x^2)) / (1-x-x^2), c(x) the g.f. of A000108.
%C A184018 Hankel transform is (9,-5) Somos-4 sequence A184019.
%C A184018 The radius of convergence r of the g.f. A(x) satisfies: r = (1-r-r^2)/4 = limit a(n)/a(n+1) = (sqrt(29)-5)/2 = 0.19258240... with A(r) = 1/(2*r) = (sqrt(29)+5)/4 = 2.59629120... - _Paul D. Hanna_, Sep 06 2011
%H A184018 Vincenzo Librandi, <a href="/A184018/b184018.txt">Table of n, a(n) for n = 0..1000</a>
%F A184018 G.f.: ( 1 - x - x^2 - sqrt((1 - x - x^2)*(1 - 5*x - x^2)) )/( 2*x*(1 - x - x^2) ).
%F A184018 G.f.: 1/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1-... (continued fraction).
%F A184018 a(n) = Sum_{k=0..n} (Sum_{j=0..n-k} binomial(k+j,k)*binomial(j,n-k-j))*A000108(k) = Sum_{k=0..n} A037027(n,k)*A000108(k).
%F A184018 G.f. satisfies A(x) = 1/(1-x-x^2) + x*A(x)^2. - _Paul D. Hanna_, Sep 06 2011
%F A184018 Conjecture: (n+1)*a(n) + 2*(1-3*n)*a(n-1) + 3*(n-1)*a(n-2) + 2*(3*n-5)*a(n-3) + (n-3)*a(n-4) = 0. - _R. J. Mathar_, Nov 15 2011
%F A184018 a(n) ~ (27+5*sqrt(29)) * sqrt(54*sqrt(29)-290) * (5+sqrt(29))^n / (sqrt(Pi) * n^(3/2) * 2^(n+5)). - _Vaclav Kotesovec_, Feb 04 2014
%p A184018 A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
%p A184018 A037027 := proc(n,m) add( binomial(m+k,m)*binomial(k,n-k-m),k=0..n-m) ; end proc:
%p A184018 A184018 := proc(n) add( A037027(n,k)*A000108(k),k=0..n) ; end proc:
%p A184018 seq(A184018(n),n=0..10) ; # _R. J. Mathar_, Jan 11 2011
%t A184018 CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 6 x^3 + x^4])/(2 x (1 - x - x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 04 2014 *)
%o A184018 (PARI) {a(n)=polcoeff((1-sqrt(1-4*x/(1-x-x^2 +O(x^(n+2)))))/(2*x),n)} /* _Paul D. Hanna_, Sep 06 2011 */
%K A184018 nonn,easy
%O A184018 0,2
%A A184018 _Paul Barry_, Jan 08 2011