A134761 a(n) = (1/2)*( (1 + (-1)^n)*A134760(n/2) + (1 - (-1)^n) ).

1, 1, 3, 1, 11, 1, 39, 1, 139, 1, 503, 1, 1847, 1, 6863, 1, 25739, 1, 97239, 1, 369511, 1, 1410863, 1, 5408311, 1, 20801199, 1, 80233199, 1, 310235039, 1, 1202160779, 1, 4667212439, 1, 18150270599, 1, 70690527599, 1, 275693057639, 1, 1076515748879, 1, 4208197927439, 1

Offset: 0

Author

Gary W. Adamson, Nov 09 2007

Keywords

• nonn

Comments

Second inverse binomial transform of A134760.

A134760 interpolated with 1's.

Former name: A007318^(-2) * A134760. - G. C. Greubel, May 27 2024

Examples

			The first few terms are (1, 1, 3, 1, 11, 1, 39, ...), since A134760 = (1, 3, 11, 39, 139, 503, ...).
		

Links

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

Crossrefs

Cf. A000984, A134760.

Programs

Magma

A134761:= func< n | (n mod 2 eq 0) select 2*Binomial(2*Floor(n/2), Floor(n/2)) - 1 else 1 >;
[A134761(n): n in [0..70]]; // G. C. Greubel, May 27 2024

Mathematica

Table[If[EvenQ[n], 2*(1+Floor[n/2])*CatalanNumber[Floor[n/2]]-1, 1], {n,0,70}] (* G. C. Greubel, May 27 2024 *)

SageMath

def A134761(n): return 1 if (n%2==0) else 2*binomial(2*(n//2), (n//2)) -1
[A134761(n) for n in range(71)] # G. C. Greubel, May 27 2024

Formula

From G. C. Greubel, May 27 2024: (Start)

a(n) = (1/2)*( (1 + (-1)^n)*A134760(n/2) + (1 - (-1)^n) ).

G.f.: 2/sqrt(1 - 4*x^2) - 1/(1 + x).

E.g.f.: 2*BesselI(0, 2*x) - exp(-x).

a(n) = (-(n-1)*(3*n-4)*a(n-1) + 4*(3*n^2 -10*n +7)*a(n-2) + 4*(n-2)*(3*n-4)*a(n-3))/(n*(3*n-7)), with a(0) = a(1) = 1, a(2) = 3. (End)

Extensions

New name and terms a(14) onward added by G. C. Greubel, May 27 2024