A239201 - OEIS

%I #22 Nov 19 2021 09:26:03

%S 2,5,17,68,293,1310,5984,27725,129773,612158,2905322,13857035,

%T 66361892,318901523,1536964313,7426185908,35960185373,174468439958,

%U 847920579938,4127211830363,20116566452918,98172213841553,479635277636543,2345731259059238,11482918774964588,56260052353307435,275862429353287079,1353641461527506630

%N Expansion of -(x * sqrt(5*x^2 -6*x +1) -2*x^3 +3*x^2 -x) / ((3*x^2 -4*x +1) * sqrt(5*x^2 -6*x +1) +5*x^3 -11*x^2 +7*x -1).

%F G.f. A(x) = G'(x)*(x*G(x)-x^2)/G(x)^2, where G(x) = A007317(x) = (sqrt(5*x^2-6*x+1)+x-1)/(2*x-2).

%F a(n) = [x^n] (F(x)^n-F(x)^(n-1)), where F(x) = (x^2-x-1)/(x-1).

%F a(n) = sum(k=1..n, binomial(n-1,n-k)*sum(i=0..n-k, binomial(k,n-k-i)*binomial(k+i-1,k-1)*2^(-n+2*k+i)*(-1)^(n-k-i))), n>0.

%F Conjecture D-finite with recurrence: (-n+1)*a(n) +(7*n-11)*a(n-1) +(-11*n+25)*a(n-2) +5*(n-3)*a(n-3)=0. - _R. J. Mathar_, Oct 07 2016

%F a(n) ~ 3 * 5^(n - 1/2) / (4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Nov 19 2021

%o (Maxima)

%o a(n):=sum(binomial(n-1,n-k)*sum(binomial(k,n-k-i)*binomial(k+i-1,k-1)*2^(-n+2*k+i)*(-1)^(n-k-i),i,0,n-k),k,1,n);

%o (PARI) x='x+O('x^66); G=(sqrt(5*x^2-6*x+1)+x-1)/(2*x-2); Vec(G' * (x * G - x^2 ) / G^2) \\ _Joerg Arndt_, Mar 12 2014

%Y Cf. A007317.

%K nonn

%O 1,1

%A _Vladimir Kruchinin_, Mar 12 2014