A098401 a(n) = (0^n + 3^n*binomial(2*n,n))/2.
1, 3, 27, 270, 2835, 30618, 336798, 3752892, 42220035, 478493730, 5454828522, 62482581252, 718549684398, 8290957896900, 95938227092700, 1112883434275320, 12937269923450595, 150681143814306930, 1757946677833580850, 20540219077844997300, 240320563210786468410
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..900
Programs
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Magma
[(0^n + 3^n * Binomial(2*n, n))/2: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012
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Mathematica
CoefficientList[Series[(6x)/(Sqrt[1-12x](1-Sqrt[1-12x])),{x,0,30}],x] (* Harvey P. Dale, Nov 29 2023 *) Table[(3^n*Binomial[2*n,n] +Boole[n==0])/2, {n,0,40}] (* G. C. Greubel, Dec 27 2023 *) a[n_] := 3^n*HypergeometricPFQ[{-n, -n + 1}, {1}, 1]; Flatten[Table[a[n], {n,0,20}]] (* Detlef Meya, May 21 2024 *) -
SageMath
[(3^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023
Formula
a(n+1) = 3*A098399(n).
G.f.: 6*x/(sqrt(1-12*x)*(1-sqrt(1-12*x))).
n*a(n) - 6*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 13/11 + 24*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 11/13 - 24*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)
a(n) = 3^n*hypergeom([-n, -n + 1], [1], 1). - Detlef Meya, May 21 2024