A144706 Central coefficients of the triangle A132047.

1, 6, 18, 60, 210, 756, 2772, 10296, 38610, 145860, 554268, 2116296, 8112468, 31201800, 120349800, 465352560, 1803241170, 7000818660, 27225405900, 106035791400, 413539586460, 1614773623320, 6312296891160, 24700292182800, 96742811049300, 379231819313256

Offset: 0

Author

Paul Barry, Sep 19 2008

Keywords

• easy
• nonn

Comments

Hankel transform is A144708.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A010686, A039599, A082505, A106566, A132047, A144708.

Programs

Magma

[n eq 0 select 1 else 3*(n+1)*Catalan(n): n in [0..40]]; // G. C. Greubel, Jun 16 2022

Mathematica

Table[3*Binomial[2n,n] -2*Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 16 2022 *)

PARI

a(n) = if(n, 3*binomial(2*n, n), 1) \\ Charles R Greathouse IV, Oct 23 2023

SageMath

[3*binomial(2*n, n) -2*bool(n==0) for n in (0..40)] # G. C. Greubel, Jun 16 2022

Formula

G.f.: 3/sqrt(1-4*x) - 2;

a(n) = 3*binomial(2*n, n) - 2*0^n.

From Philippe Deléham, Oct 30 2008: (Start)

a(n) = Sum_{k=0..n} A039599(n,k)*A010686(k).

a(n) = Sum_{k=0..n} A106566(n,k)*A082505(k+1). (End)

D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1). - R. J. Mathar, Nov 30 2012

E.g.f.: -2 + 3*exp(2*x)*BesselI(0, 2*x). - G. C. Greubel, Jun 16 2022