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A373614
a(n) = Fibonacci(n)^2 * Catalan(n).
0
0, 1, 2, 20, 126, 1050, 8448, 72501, 630630, 5620472, 50807900, 465643906, 4313336832, 40331298100, 380115482760, 3607451824500, 34444346026230, 330647239219110, 3189220347667200, 30893105448487590, 300408447948394500, 2931423727834870320, 28696206742447216440, 281728667746183208850, 2773282854528632549376
OFFSET
0,3
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).
a(n) = A007598(n)*A000108(n).
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