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A037966 a(n) = n^2*binomial(2*n-2, n-1).

%I A037966 #50 Dec 28 2022 01:52:49
%S A037966 0,1,8,54,320,1750,9072,45276,219648,1042470,4862000,22355476,
%T A037966 101582208,457002364,2038517600,9026235000,39710085120,173712232710,
%U A037966 756088415280,3276123843300,14138105520000,60790319209620,260516811228960,1113068351807880,4742456099097600,20154752301937500,85453569951920352
%N A037966 a(n) = n^2*binomial(2*n-2, n-1).
%D A037966 The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972.
%H A037966 Seiichi Manyama, <a href="/A037966/b037966.txt">Table of n, a(n) for n = 0..1000</a>
%H A037966 Nikita Gogin and Mika Hirvensalo, <a href="https://pca-pdmi.ru/2020/files/10/GoHi2020ExtAbstract.pdf">On the Moments of Squared Binomial Coefficients</a>, (2020).
%H A037966 Han Mao Kiah, Alexander Vardy, and Hanwen Yao, <a href="https://arxiv.org/abs/2212.09952">Efficient Algorithms for the Bee-Identification Problem</a>, arXiv:2212.09952 [cs.IT], 2022.
%F A037966 a(n) = Sum_{k=0..n} k^2*binomial(n,k)^2. - _Paul Barry_, Mar 04 2003
%F A037966 a(n) = n^2*A000984(n-1). - _Zerinvary Lajos_, Jan 18 2007, corrected Jul 26 2015
%F A037966 a(n) = n*A037965(n). - _Zerinvary Lajos_, Jan 18 2007, corrected Jul 26 2015
%F A037966 (n-1)^3*a(n) = 2*n^2*(2*n-3)*a(n-1). - _R. J. Mathar_, Jul 26 2015
%F A037966 E.g.f.: x*exp(2*x)*((1 + 2*x)*BesselI(0,2*x) + 2*x*BesselI(1,2*x)). - _Ilya Gutkovskiy_, Mar 04 2021
%t A037966 Array[#^2*Binomial[2#-2, #-1] &, 27, 0] (* _Michael De Vlieger_, Jul 15 2020 *)
%o A037966 (PARI) {a(n) = n^2*binomial(2*n-2, n-1)} \\ _Seiichi Manyama_, Jul 15 2020
%o A037966 (Magma) [0] cat [n^3*Catalan(n-1): n in [1..30]]; // _G. C. Greubel_, Jun 19 2022
%o A037966 (SageMath) [n^3*catalan_number(n-1) for n in (0..30)] # _G. C. Greubel_, Jun 19 2022
%Y A037966 Cf. A000108, A000984, A037965, A336214.
%K A037966 nonn,easy
%O A037966 0,3
%A A037966 _N. J. A. Sloane_
%E A037966 More terms from _Seiichi Manyama_, Jul 15 2020