OFFSET
0,2
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.
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)
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 *)
PROG
(Magma) [2*(n+1)*Catalan(n)-1: n in [0..40]]; // G. C. Greubel, Apr 06 2024
(SageMath) [2*binomial(2*n, n)-1 for n in range(41)] # G. C. Greubel, Apr 06 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Nov 09 2007
STATUS
approved