A099044 a(n) = (2*0^n + 3^n*binomial(2*n,n))/3.

1, 2, 18, 180, 1890, 20412, 224532, 2501928, 28146690, 318995820, 3636552348, 41655054168, 479033122932, 5527305264600, 63958818061800, 741922289516880, 8624846615633730, 100454095876204620, 1171964451889053900, 13693479385229998200, 160213708807190978940

Offset

0,2

Comments

(1 + (k-1)*sqrt(1-4*k*x))/(k*sqrt(1-4*k*x)) is the g.f. for ((k-1)*0^n + k^n*binomial(2*n,n))/k.

Links

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

Formula

G.f.: 1/3 + 4*x/(sqrt(1-12*x)*(1-sqrt(1-12*x))) = (1 + 2*sqrt(1-12*x))/(3*sqrt(1-12*x)).

n*a(n) + 6*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012

E.g.f.: (2 + exp(6*x) * BesselI(0,6*x)) / 3. - Ilya Gutkovskiy, Nov 17 2021

From Amiram Eldar, Feb 17 2026: (Start)

Sum_{n>=0} 1/a(n) = 14/11 + 36*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).

Sum_{n>=0} (-1)^n/a(n) = 10/13 - 36*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)

Mathematica

Join[{1}, Table[3^(n-1)*Binomial[2*n, n], {n, 1, 30}]] (* G. C. Greubel, Dec 31 2017 *)

Prog

(Magma) [(2*0^n + 3^n*Binomial(2*n, n))/3: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012
(PARI) a(n) = (2*0^n + 3^n*binomial(2*n, n))/3; \\ G. C. Greubel, Dec 31 2017

Crossrefs

Cf. A069723, A088218, A099045, A099046.

Sequence in context: A264196 A214768 A337855 * A161122 A019581 A183285

Adjacent sequences: A099041 A099042 A099043 * A099045 A099046 A099047

Keyword

easy,nonn

Author

Paul Barry, Sep 24 2004

Status

approved