A373614 - OEIS

%I #24 Jun 12 2024 13:37:16

%S 0,1,2,20,126,1050,8448,72501,630630,5620472,50807900,465643906,

%T 4313336832,40331298100,380115482760,3607451824500,34444346026230,

%U 330647239219110,3189220347667200,30893105448487590,300408447948394500,2931423727834870320,28696206742447216440,281728667746183208850,2773282854528632549376

%N a(n) = Fibonacci(n)^2 * Catalan(n).

%H Vladimir V. Kruchinin and Maria Y. Perminova, <a href="https://arxiv.org/abs/2406.02937">Identities and Generating Functions of Products of Generalized Fibonacci numbers, Catalan and Harmonic Numbers</a>, arXiv:2406.02937 [math.CO], 2024.

%F 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).

%F a(n) = A007598(n)*A000108(n).

%F 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

%p 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):

%p seq(coeff(ser, x, n), n = 0..24); # _Peter Luschny_, Jun 11 2024

%t 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 *)

%t (* A variant that does not need assumptions: *)

%t 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);

%t Round[CoefficientList[Series[gf, {x, 0, 24}], x]]  (* _Peter Luschny_, Jun 11 2024 *)

%Y Cf. A000045, A000108, A007598, A119694.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Jun 10 2024