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%I #25 Jan 13 2023 18:58:11
%S 1,5,53,662,8885,124130,1778966,25947612,383358645,5719519850,
%T 85990654178,1300866635172,19780031677718,302045506654052,
%U 4629016098160220,71163013287905912,1096960888092571317,16949379732631632570,262435310495071434602
%N a(n) = (binomial(4*n, 2*n) + binomial(2*n, n)^2)/2.
%D The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972, Eq 3.68, page 30.
%H G. C. Greubel, <a href="/A036910/b036910.txt">Table of n, a(n) for n = 0..500</a>
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 6.
%F a(n) = Sum_{k=0..n} binomial(2n, k)^2. - _Paul Barry_, May 15 2003
%F From _G. C. Greubel_, Dec 09 2021: (Start)
%F a(n) = A000984(2*n) + A000984(n)^2.
%F a(n) = A001448(n) + A000984(n)^2.
%F a(n) = A157531(2n, n)/2. - corrected Jan 12 2023
%F G.f.: sqrt(1 + sqrt(1 - 16*x))/(2*sqrt(2)*sqrt(1 - 16*x)) + (1/Pi)*EllipticK[16*x]. (End)
%F D-finite with recurrence n^2*(2*n-1)*(n-1)*a(n) -2*(n-1) *(2*n^3+109*n^2-241*n+132) *a(n-1) +12*(-112*n^4+1008*n^3-3011*n^2+3702*n-1600) *a(n-2) +16*(832*n^4-7008*n^3+20744*n^2-24228*n+7905) *a(n-3) +256*(4*n-13) *(4*n-15)*(-7+2*n)^2*a(n-4)=0. - _R. J. Mathar_, Jan 12 2023
%t B[n_] := Binomial[2*n, n]/2; Table[B[2*n] + 2*B[n]^2, {n, 0, 40}] (* _G. C. Greubel_, Dec 09 2021 *)
%o (Magma) [(Binomial(4*n, 2*n) + Binomial(2*n, n)^2)/2: n in [0..40]]; // G. C. Greubel, Dec 09 2021
%o (Sage) [(binomial(4*n, 2*n) + binomial(2*n, n)^2)/2 for n in (0..40)] # G. C. Greubel, Dec 09 2021
%o (PARI) a(n) = (binomial(4*n,2*n)+binomial(2*n,n)^2)/2; \\ _Michel Marcus_, Dec 09 2021
%Y Cf. A000984, A001448, A157531.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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