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

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..900

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)

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 *)

Prog

(Magma) [(0^n + 3^n * Binomial(2*n, n))/2: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012
(SageMath) [(3^n*binomial(2*n, n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023

Crossrefs

Cf. A069723, A069720, A098399, A098400, A098402.

Sequence in context: A026294 A199688 A275998 * A355363 A011544 A127503

Adjacent sequences: A098398 A098399 A098400 * A098402 A098403 A098404

Keyword

easy,nonn

Author

Paul Barry, Sep 06 2004

Status

approved