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

%I A134760 #24 Apr 08 2024 09:25:52
%S A134760 1,3,11,39,139,503,1847,6863,25739,97239,369511,1410863,5408311,
%T A134760 20801199,80233199,310235039,1202160779,4667212439,18150270599,
%U A134760 70690527599,275693057639,1076515748879,4208197927439,16466861455199,64495207366199,252821212875503
%N A134760 a(n) = 2*A000984(n) - 1.
%C A134760 Inverse binomial transform of this is A134761: (the sequence interpolated with ones): (1, 1, 3, 1, 11, 1, 39, 1, 139, ...).
%H A134760 Alois P. Heinz, <a href="/A134760/b134760.txt">Table of n, a(n) for n = 0..500</a>
%H A134760 C. J. Fewster and D. Siemssen, <a href="http://arxiv.org/abs/1403.1723">Enumerating Permutations by their Run Structure</a>, arXiv preprint arXiv:1403.1723 [math.CO], 2014.
%F A134760 From _R. J. Mathar_, Mar 23 2015: (Start)
%F A134760 n*a(n) = 2*(3*n-2)*a(n-1) - (9*n-14)*a(n-2) + 2*(2*n-5)*a(n-3).
%F A134760 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)
%F A134760 From _G. C. Greubel_, Apr 06 2024: (Start)
%F A134760 a(n) = 2*(n+1)*A000108(n) - 1.
%F A134760 G.f.: 2/sqrt(1 - 4*x) - 1/(1 - x).
%F A134760 E.g.f.: 2*exp(2*x)*BesselI(0, 2*x) - exp(x). (End)
%p A134760 a:= proc(n) option remember; `if`(n<2, 2*n+1,
%p A134760        ((12-31*n+15*n^2) *a(n-1)
%p A134760         -2*(3*n-2)*(2*n-3)*a(n-2)) / (n*(3*n-5)))
%p A134760     end:
%p A134760 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 16 2013
%t A134760 a[n_] := 2 Binomial[2n, n] - 1; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jul 21 2017 *)
%o A134760 (Magma) [2*(n+1)*Catalan(n)-1: n in [0..40]]; // _G. C. Greubel_, Apr 06 2024
%o A134760 (SageMath) [2*binomial(2*n,n)-1 for n in range(41)] # _G. C. Greubel_, Apr 06 2024
%Y A134760 Cf. A000984, A000108, A134761.
%K A134760 nonn,easy
%O A134760 0,2
%A A134760 _Gary W. Adamson_, Nov 09 2007