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%I #29 Sep 08 2022 08:44:53
%S 1,3,21,210,2485,31878,427350,5891028,82824885,1181976510,17067482146,
%T 248817506028,3656231188246,54086245380300,804670817838300,
%U 12030722583033960,180648817921816245,2722858996178147310,41179040361190612650,624643836563467851900
%N a(n) = (binomial(2*n, n)^2 + binomial(2*n, n))/2.
%D The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972, Eq. (3.82), page 31.
%H Alois P. Heinz, <a href="/A037967/b037967.txt">Table of n, a(n) for n = 0..833</a> (first 101 terms from Chai Wah Wu)
%F a(n) = (-1)^n*Sum_{k=0..n} (-1)^k*binomial(2*n, k)^2.
%F n^2*(n-1)*(3*n-5)*a(n) - 2*(n-1)*(2*n-1)*(15*n^2-31*n+12)*a(n-1) + 8*(2*n-1)*(3*n-2)*(2*n-3)^2*a(n-2) = 0. - _R. J. Mathar_, Jul 26 2015
%F a(n) = A000217(A000984(n)). - _Alois P. Heinz_, Aug 10 2020
%p a:= n-> (h-> h*(h+1)/2)(binomial(2*n, n)):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 10 2020
%t Table[(Binomial[2n, n]^2 + Binomial[2n, n])/2, {n,0,45}] (* _Vincenzo Librandi_, Jun 02 2015 *)
%o (Python)
%o from gmpy2 import bincoef
%o def A037967(n):
%o return bincoef(bincoef(2*n,n)+1,2) # _Chai Wah Wu_, Jun 01 2015
%o (Magma) [(Binomial(2*n,n)^2+Binomial(2*n,n))/2: n in [0..30]]; // _Vincenzo Librandi_, Jun 02 2015
%o (PARI) a(n)=binomial(binomial(2*n,n)+1,2) \\ _Charles R Greathouse IV_, Jun 02 2015
%o (SageMath) [binomial(1+(n+1)*catalan_number(n), 2) for n in (0..30)] # _G. C. Greubel_, Jun 19 2022
%Y Cf. A000108, A000217, A000984.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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