OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: -(2*x+sqrt(1-4*x)-1)/((2*sqrt(1-4*x)*x-8*x+2)*x^2).
a(n) ~ 2^(2*n+4)/sqrt(Pi*n). - Vaclav Kotesovec, Nov 23 2016
Conjecture: +(n+2)*(n^2-4*n+1)*a(n) +2*(-4*n^3+9*n^2+16*n-6)*a(n-1) +(15*n^3-38*n^2-9*n+14)*a(n-2) +2*(2*n-1)*(n^2-2*n-2)*a(n-3)=0. - R. J. Mathar, Mar 06 2017
Conjecture: +(n+2)*a(n) +(-11*n-14)*a(n-1) +(37*n+20)*a(n-2) +(-25*n-6)*a(n-3) +2*(-21*n+41)*a(n-4) +4*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Mar 06 2017
MATHEMATICA
Table[Sum[Fibonacci[j+1]*Binomial[2*n-j+2, n+2], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 23 2016 *)
FullSimplify[Table[Binomial[2*n+2, n+2]*(Hypergeometric2F1[1, -n, -2*(n+1), -1/GoldenRatio] + GoldenRatio^2 * Hypergeometric2F1[1, -n, -2*(n+1), GoldenRatio])/(Sqrt[5]*GoldenRatio), {n, 0, 20}]] (* Vaclav Kotesovec, Nov 23 2016 *)
PROG
(Maxima)
taylor(-(2*x+sqrt(1-4*x)-1)/(2*sqrt(1-4*x)*x-8*x+2)/x^2, x, 0, 20)
(PARI) x='x+O('x^50); Vec(-(2*x+sqrt(1-4*x)-1)/((2*sqrt(1-4*x)*x-8*x+2)*x^2)) \\ G. C. Greubel, Jun 07 2017
(PARI) a(n) = sum(i=0, n, fibonacci(i+1)*binomial(2*n-i+2, n+2)); \\ Michel Marcus, Jun 06 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Nov 23 2016
STATUS
approved