A134760 a(n) = 2*A000984(n) - 1.

1, 3, 11, 39, 139, 503, 1847, 6863, 25739, 97239, 369511, 1410863, 5408311, 20801199, 80233199, 310235039, 1202160779, 4667212439, 18150270599, 70690527599, 275693057639, 1076515748879, 4208197927439, 16466861455199, 64495207366199, 252821212875503

Offset: 0

Author

Gary W. Adamson, Nov 09 2007

Keywords

• nonn
• easy

Comments

Inverse binomial transform of this is A134761: (the sequence interpolated with ones): (1, 1, 3, 1, 11, 1, 39, 1, 139, ...).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• C. J. Fewster and D. Siemssen, Enumerating Permutations by their Run Structure, arXiv preprint arXiv:1403.1723 [math.CO], 2014.

Crossrefs

Cf. A000984, A000108, A134761.

Programs

Magma

[2*(n+1)*Catalan(n)-1: n in [0..40]]; // G. C. Greubel, Apr 06 2024

Maple

a:= proc(n) option remember; `if`(n<2, 2*n+1,
       ((12-31*n+15*n^2) *a(n-1)
        -2*(3*n-2)*(2*n-3)*a(n-2)) / (n*(3*n-5)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 16 2013

Mathematica

a[n_] := 2 Binomial[2n, n] - 1; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 21 2017 *)

SageMath

[2*binomial(2*n,n)-1 for n in range(41)] # G. C. Greubel, Apr 06 2024

Formula

From R. J. Mathar, Mar 23 2015: (Start)

n*a(n) = 2*(3*n-2)*a(n-1) - (9*n-14)*a(n-2) + 2*(2*n-5)*a(n-3).

n*(3*n-5)*a(n) = (15*n^2-31*n+12)*a(n-1) - 2*(3*n-2)*(2*n-3)*a(n-2). (End)

From G. C. Greubel, Apr 06 2024: (Start)

a(n) = 2*(n+1)*A000108(n) - 1.

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

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