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

%I A134761 #8 May 27 2024 22:47:26
%S A134761 1,1,3,1,11,1,39,1,139,1,503,1,1847,1,6863,1,25739,1,97239,1,369511,1,
%T A134761 1410863,1,5408311,1,20801199,1,80233199,1,310235039,1,1202160779,1,
%U A134761 4667212439,1,18150270599,1,70690527599,1,275693057639,1,1076515748879,1,4208197927439,1
%N A134761 a(n) = (1/2)*( (1 + (-1)^n)*A134760(n/2) + (1 - (-1)^n) ).
%C A134761 Second inverse binomial transform of A134760.
%C A134761 A134760 interpolated with 1's.
%C A134761 Former name: A007318^(-2) * A134760. - _G. C. Greubel_, May 27 2024
%H A134761 G. C. Greubel, <a href="/A134761/b134761.txt">Table of n, a(n) for n = 0..1000</a>
%F A134761 From _G. C. Greubel_, May 27 2024: (Start)
%F A134761 a(n) = (1/2)*( (1 + (-1)^n)*A134760(n/2) + (1 - (-1)^n) ).
%F A134761 G.f.: 2/sqrt(1 - 4*x^2) - 1/(1 + x).
%F A134761 E.g.f.: 2*BesselI(0, 2*x) - exp(-x).
%F A134761 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)
%e A134761 The first few terms are (1, 1, 3, 1, 11, 1, 39, ...), since A134760 = (1, 3, 11, 39, 139, 503, ...).
%t A134761 Table[If[EvenQ[n], 2*(1+Floor[n/2])*CatalanNumber[Floor[n/2]]-1, 1], {n,0,70}] (* _G. C. Greubel_, May 27 2024 *)
%o A134761 (Magma)
%o A134761 A134761:= func< n | (n mod 2 eq 0) select 2*Binomial(2*Floor(n/2), Floor(n/2)) - 1 else 1 >;
%o A134761 [A134761(n): n in [0..70]]; // _G. C. Greubel_, May 27 2024
%o A134761 (SageMath)
%o A134761 def A134761(n): return 1 if (n%2==0) else 2*binomial(2*(n//2), (n//2)) -1
%o A134761 [A134761(n) for n in range(71)] # _G. C. Greubel_, May 27 2024
%Y A134761 Cf. A000984, A134760.
%K A134761 nonn
%O A134761 0,3
%A A134761 _Gary W. Adamson_, Nov 09 2007
%E A134761 New name and terms a(14) onward added by _G. C. Greubel_, May 27 2024