OFFSET
0,3
LINKS
Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Generating Functions of Products of Generalized Fibonacci numbers, Catalan and Harmonic Numbers, arXiv:2406.02937 [math.CO], 2024.
FORMULA
G.f.: (2*sqrt(-sqrt(16*x^2 - 12*x + 1) - 6*x + 1)/(5*sqrt(10)*x) + 3*(1 -sqrt((-2*sqrt(16*x^2 - 12*x + 1) - 42*x + 7)/5 + 6*x))/(10*x)) + (1 -sqrt(4*x + 1))/(5*x).
D-finite with recurrence n*(n-1)*(n+1)*a(n) -4*n*(2*n-1)*(n-1)*a(n-1) -8*(n-1)*(2*n-1)*(2*n-3)*a(n-2) +8*(2*n-5)*(2*n-1)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jun 12 2024
MAPLE
gf := (2 * sqrt(-sqrt(16*x^2 - 12*x+1) - 6*x + 1) / (5*sqrt(10)*x) + 3*(1 -sqrt((-2*sqrt(16*x^2 - 12*x + 1) - 42*x + 7) / 5 + 6*x))/(10*x)) + (1 -sqrt(4*x + 1)) / (5*x): assume(x > 0): ser := series(gf, x, 30):
seq(coeff(ser, x, n), n = 0..24); # Peter Luschny, Jun 11 2024
MATHEMATICA
CoefficientList[Series[(2*Sqrt[-Sqrt[16*x^2 - 12*x + 1] - 6*x + 1]/(5*Sqrt[10]*x) + 3*(1 -Sqrt[(-2*Sqrt[16*x^2 - 12*x + 1] - 42*x + 7)/5 + 6*x])/(10*x)) + (1 -Sqrt[4*x + 1])/(5*x), {x, 0, 24}, Assumptions->(x>0)], x] (* Stefano Spezia, Jun 11 2024 *)
(* A variant that does not need assumptions: *)
gf := ((2 Sqrt[1 - 2 x (Sqrt[5] + 3)] + Sqrt[2] (Sqrt[5] + 2) Sqrt[3 + Sqrt[5] - 8 x] + (Sqrt[5] + 3) (2 Sqrt[4 x + 1] - 5)) (Sqrt[5] - 3)) / (40 x);
Round[CoefficientList[Series[gf, {x, 0, 24}], x]] (* Peter Luschny, Jun 11 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Jun 10 2024
STATUS
approved