A134770 a(n) = 4*A000984(n) - 3.

1, 5, 21, 77, 277, 1005, 3693, 13725, 51477, 194477, 739021, 2821725, 10816621, 41602397, 160466397, 620470077, 2404321557, 9334424877, 36300541197, 141381055197, 551386115277, 2153031497757, 8416395854877, 32933722910397, 128990414732397, 505642425751005, 1983674131792413

Offset: 0

Author

Gary W. Adamson, Nov 10 2007

Keywords

• nonn

Comments

The second inverse binomial transform of this sequence is A134771, the sequence interleaved with threes: (1, 3, 5, 3, 21, 3, 77, 3, ...).

Examples

			a(2) = 21 = 4*A000984(2) - 3 = 4*6 - 3.
		

Links

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

Crossrefs

Cf. A000108, A000984, A134771.

Programs

Magma

[4*(n+1)*Catalan(n)-3: n in [0..40]]; // G. C. Greubel, Oct 13 2023

Mathematica

Table[4 Binomial[2n,n]-3,{n,0,30}] (* Harvey P. Dale, Dec 01 2022 *)

PARI

a(n)=4*binomial(2*n, n) - 3; \\ Michel Marcus, Jul 02 2020

SageMath

[4*binomial(2*n,n)-3 for n in range(41)] # G. C. Greubel, Oct 13 2023

Formula

From G. C. Greubel, Oct 13 2023: (Start)

a(n) = 4*(n+1)*A000108(n) - 3.

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

Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = 4*BesselI(0, 2*x) - cosh(x). (End)

Extensions

a(10) corrected and offsets aligned by Georg Fischer, Jul 01 2020

More terms from Michel Marcus, Jul 02 2020