A239201 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).

2, 5, 17, 68, 293, 1310, 5984, 27725, 129773, 612158, 2905322, 13857035, 66361892, 318901523, 1536964313, 7426185908, 35960185373, 174468439958, 847920579938, 4127211830363, 20116566452918, 98172213841553, 479635277636543, 2345731259059238, 11482918774964588, 56260052353307435, 275862429353287079, 1353641461527506630

Offset: 1

Author

Vladimir Kruchinin, Mar 12 2014

Keywords

• nonn

Crossrefs

Cf. A007317.

Programs

Maxima

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);

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

Formula

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).

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

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.

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

a(n) ~ 3 * 5^(n - 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 19 2021