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

0,2

Comments

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

Links

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

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)

Example

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

Mathematica

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

Prog

(PARI) a(n)=4*binomial(2*n, n) - 3; \\ Michel Marcus, Jul 02 2020
(Magma) [4*(n+1)*Catalan(n)-3: n in [0..40]]; // G. C. Greubel, Oct 13 2023
(SageMath) [4*binomial(2*n, n)-3 for n in range(41)] # G. C. Greubel, Oct 13 2023

Crossrefs

Cf. A000108, A000984, A134771.

Sequence in context: A026329 A014533 A255453 * A272787 A084780 A268202

Adjacent sequences: A134767 A134768 A134769 * A134771 A134772 A134773

Keyword

nonn

Author

Gary W. Adamson, Nov 10 2007

Extensions

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

More terms from Michel Marcus, Jul 02 2020

Status

approved