A098400 a(n) = 4^n*binomial(2*n+1, n).

1, 12, 160, 2240, 32256, 473088, 7028736, 105431040, 1593180160, 24216338432, 369849532416, 5671026163712, 87246556364800, 1346089726771200, 20819521107394560, 322702577164615680, 5011381198321090560, 77954818640550297600, 1214454016715941478400

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..820

Formula

G.f.: (1-sqrt(1-16*x))/(8*x*sqrt(1-16*x)).

E.g.f.: a(n) = n! * [x^n] exp(8*x)*(BesselI(0, 8*x) + BesselI(1, 8*x)). - Peter Luschny, Aug 25 2012

(n+1)*a(n) - 8*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 26 2012

a(n) = 4^n*(2*n+1)*Hypergeometric2F1([1-n,-n],[2],1). - Peter Luschny, Sep 22 2014

From G. C. Greubel, Dec 27 2023: (Start)

a(n) = 4^n * A001700(n).

a(n) = 4^n * (2*n+1) * A000108(n).

a(n) = (2*n+1) * A151403(n). (End)

From Amiram Eldar, Jan 16 2024: (Start)

Sum_{n>=0} 1/a(n) = 8/15 + 128*arcsin(1/4)/(15*sqrt(15)).

Sum_{n>=0} (-1)^n/a(n) = 8/17 + 128*arcsinh(1/4)/(17*sqrt(17)). (End)

Mathematica

Table[4^n Binomial[2n+1, n], {n, 0, 20}] (* Harvey P. Dale, Jan 22 2019 *)

Prog

(PARI) a(n)=binomial(2*n+1, n)<<(2*n) \\ Charles R Greathouse IV, Oct 23 2023
(Magma) [4^n*(2*n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Dec 27 2023
(SageMath) [4^n*binomial(2*n+1, n) for n in range(31)] # G. C. Greubel, Dec 27 2023

Crossrefs

Cf. A000108, A001700, A069720, A069723, A098399, A151403.

Sequence in context: A091019 A355844 A219418 * A208791 A019578 A144346

Adjacent sequences: A098397 A098398 A098399 * A098401 A098402 A098403

Keyword

easy,nonn

Author

Paul Barry, Sep 06 2004

Status

approved