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%I #31 Sep 08 2022 08:45:21
%S 1,5,46,517,6376,82994,1119210,15475205,217994860,3115374880,
%T 45035696036,657153097330,9663914317396,143050882063262,
%U 2129448324373546,31853280798384645,478503774600509620,7215090439396842572,109154411037070011504,1656268648035559711392
%N a(n) = Sum_{k=0..n} binomial(n+k, k)^2.
%H Vincenzo Librandi, <a href="/A112029/b112029.txt">Table of n, a(n) for n = 0..200</a>
%H C. Elsner, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-5.pdf">On recurrence formulas for sums involving binomial coefficients</a>, Fib. Q., 43,1 (2005), 31-45.
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012
%H Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, <a href="http://arxiv.org/abs/1602.04347">Sums of powers of Catalan triangle numbers</a>, arXiv:1602.04347 [math.NT], 2016.
%F a(n) ~ 2^(4*n+2)/(3*Pi*n). - _Vaclav Kotesovec_, Nov 23 2012
%F Recurrence: 2*(2*n+1)*(21*n-13)*n^2*a(n) = (1365*n^4 - 1517*n^3 + 240*n^2 + 216*n - 64)*a(n-1) - 4*(n-1)*(2*n-1)^2*(21*n+8)*a(n-2). - _Vaclav Kotesovec_, Nov 23 2012
%F G.f.: see Maple code. - _Mark van Hoeij_, Mar 27 2013
%p f := 64*x^2/(16*x-1); S := sqrt(x)*sqrt(4-x);
%p H := ((10*x-5/8)*hypergeom([1/4,1/4],[1],f)-(21*x-21/8)*hypergeom([1/4,5/4],[1],f))/(S*(1-16*x)^(5/4));
%p ord := 30;
%p ogf := series(int(series(H,x=0,ord),x)/S,x=0,ord);
%p # _Mark van Hoeij_, Mar 27 2013
%t Table[Sum[Binomial[n+k,k]^2, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Nov 23 2012 *)
%o (Magma) [(&+[Binomial(n+j, j)^2: j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Jul 06 2021
%o (Sage) [sum(binomial(n+j, j)^2 for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Jul 06 2021
%o (PARI) a(n) = sum(k=0, n, binomial(n+k, k)^2); \\ _Michel Marcus_, Jul 07 2021
%Y Cf. A001700, A112028, A219562, A219563, A219564.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 28 2005
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