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A037965 a(n) = n*binomial(2*n-2, n-1).

%I A037965 #49 Jun 20 2022 04:41:06
%S A037965 0,1,4,18,80,350,1512,6468,27456,115830,486200,2032316,8465184,
%T A037965 35154028,145608400,601749000,2481880320,10218366630,42004911960,
%U A037965 172427570700,706905276000,2894777105220,11841673237680,48394276165560,197602337462400,806190092077500
%N A037965 a(n) = n*binomial(2*n-2, n-1).
%C A037965 a(n+1) is the convolution of A000984 and A081294. - _Paul Barry_, Sep 18 2008
%D A037965 The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972.
%H A037965 G. C. Greubel, <a href="/A037965/b037965.txt">Table of n, a(n) for n = 0..1000</a>
%F A037965 Assuming offset -1 here and offset 0 in A134757, A134757 is the inverse binomial transform of this sequence. - _Gary W. Adamson_, Nov 08 2007
%F A037965 G.f.: Hypergeometric2F1([1/2, 2], [1], 4*x). - _Paul Barry_, Sep 03 2008
%F A037965 From _Paul Barry_, Sep 18 2008: (Start)
%F A037965 G.f.: x*(1-2*x)/(1-4*x)^(3/2);
%F A037965 a(n+1) = Sum_{k=0..n} binomial(2*k,k)*(4^(n-k) + 0^(n-k))/2. (End)
%F A037965 D-finite with recurrence (n-1)*a(n) - 2*(3*n-4)*a(n-1) + 4*(2*n-5)*a(n-2) = 0. - _R. J. Mathar_, Nov 30 2012
%F A037965 E.g.f.: x*exp(2*x)*BesselI(0,2*x). - _Ilya Gutkovskiy_, Aug 22 2018
%F A037965 a(n) = n*A000984(n-1). - _R. J. Mathar_, Nov 08 2021
%F A037965 From _Amiram Eldar_, Mar 10 2022: (Start)
%F A037965 Sum_{n>=1} 1/a(n) = 4*Pi/(3*sqrt(3)) - Pi^2/9.
%F A037965 Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)/sqrt(5) - 4*log(phi)^2, where phi is the golden ratio (A001622). (End)
%t A037965 a[n_]:= n*Binomial[2*n-2, n-1]; Array[a, 30, 0] (* _Amiram Eldar_, Mar 10 2022 *)
%o A037965 (PARI) a(n) = n*binomial(2*n-2, n-1); \\ _Joerg Arndt_, Sep 04 2017
%o A037965 (Magma) [0] cat [n^2*Catalan(n-1): n in [1..30]]; // _G. C. Greubel_, Jun 19 2022
%o A037965 (SageMath) [n^2*catalan_number(n-1) for n in (0..30)] # _G. C. Greubel_, Jun 19 2022
%Y A037965 Cf. A000984, A001622, A081294, A109188 (inverse binomial transform).
%Y A037965 Cf. A000108, A100071, A134757.
%K A037965 nonn,easy
%O A037965 0,3
%A A037965 _N. J. A. Sloane_
%E A037965 More terms from _Zerinvary Lajos_, Oct 02 2007