A099045 a(n) = (3*0^n + 4^n*binomial(2*n,n))/4.

1, 2, 24, 320, 4480, 64512, 946176, 14057472, 210862080, 3186360320, 48432676864, 739699064832, 11342052327424, 174493112729600, 2692179453542400, 41639042214789120, 645405154329231360, 10022762396642181120, 155909637281100595200, 2428908033431882956800, 37890965321537374126080

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..800

Formula

G.f.: (1+3*sqrt(1-16*x))/(4*sqrt(1-16*x)).

n*a(n) + 8*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012

E.g.f.: (3 + exp(8*x) * BesselI(0,8*x)) / 4. - Ilya Gutkovskiy, Nov 17 2021

From Amiram Eldar, Feb 17 2026: (Start)

Sum_{n>=0} 1/a(n) = 19/15 + 64*arcsin(1/4)/(15*sqrt(15)).

Sum_{n>=0} (-1)^n/a(n) = 13/17 - 64*arcsinh(1/4)/(17*sqrt(17)). (End)

Mathematica

Join[{1}, Table[4^(n-1)*Binomial[2*n, n], {n, 1, 30}]] (* G. C. Greubel, Dec 31 2017 *)

Prog

(Magma) [(3*0^n + 4^n*Binomial(2*n, n))/4: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012
(PARI) a(n) = (3*0^n + 4^n*binomial(2*n, n))/4; \\ G. C. Greubel, Dec 31 2017

Crossrefs

Cf. A069723, A088218, A099044, A099046.

Sequence in context: A246610 A119491 A001864 * A181174 A209290 A333715

Adjacent sequences: A099042 A099043 A099044 * A099046 A099047 A099048

Keyword

easy,nonn

Author

Paul Barry, Sep 24 2004

Status

approved