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A144706 Central coefficients of the triangle A132047.

%I A144706 #25 Jun 16 2025 08:39:04
%S A144706 1,6,18,60,210,756,2772,10296,38610,145860,554268,2116296,8112468,
%T A144706 31201800,120349800,465352560,1803241170,7000818660,27225405900,
%U A144706 106035791400,413539586460,1614773623320,6312296891160,24700292182800,96742811049300,379231819313256
%N A144706 Central coefficients of the triangle A132047.
%C A144706 Hankel transform is A144708.
%H A144706 G. C. Greubel, <a href="/A144706/b144706.txt">Table of n, a(n) for n = 0..1000</a>
%F A144706 G.f.: 3/sqrt(1-4*x) - 2;
%F A144706 a(n) = 3*binomial(2*n, n) - 2*0^n.
%F A144706 From _Philippe Deléham_, Oct 30 2008: (Start)
%F A144706 a(n) = Sum_{k=0..n} A039599(n,k)*A010686(k).
%F A144706 a(n) = Sum_{k=0..n} A106566(n,k)*A082505(k+1). (End)
%F A144706 D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1). - _R. J. Mathar_, Nov 30 2012
%F A144706 E.g.f.: -2 + 3*exp(2*x)*BesselI(0, 2*x). - _G. C. Greubel_, Jun 16 2022
%t A144706 Table[3*Binomial[2n,n] -2*Boole[n==0], {n,0,40}] (* _G. C. Greubel_, Jun 16 2022 *)
%o A144706 (Magma) [n eq 0 select 1 else 3*(n+1)*Catalan(n): n in [0..40]]; // _G. C. Greubel_, Jun 16 2022
%o A144706 (SageMath) [3*binomial(2*n, n) -2*bool(n==0) for n in (0..40)] # _G. C. Greubel_, Jun 16 2022
%o A144706 (PARI) a(n) = if(n, 3*binomial(2*n, n), 1) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A144706 Cf. A000108, A010686, A039599, A082505, A106566, A132047, A144708.
%K A144706 easy,nonn
%O A144706 0,2
%A A144706 _Paul Barry_, Sep 19 2008