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%I #16 Jun 16 2016 23:27:29
%S 1,9,325,17577,1152501,84505509,6664647781,553268669865,
%T 47710914870133,4236909872278509,385139801423145825,
%U 35681384898462925125,3358273513450241419125,320308335005997679093125,30900030366269721747776325,3010365811746267487293617577
%N a(n) = Sum_{k=0..n} C(n,k)^2*C(n+k,k)^3, where C := binomial.
%H Vincenzo Librandi, <a href="/A111968/b111968.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 V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012
%F a(n) ~ (1+r)^(6*n+7/2)/r^(5*n+9/2)/(4*Pi^2*n^2)*sqrt((1-r)/(5-r)), where r is positive real root of the equation (1-r)^2*(1+r)^3=r^5, r = 0.77591859532439... - _Vaclav Kotesovec_, Nov 04 2012
%F Recurrence: (n-1)^2*n^4*(843719*n^6 - 10346391*n^5 + 52472779*n^4 - 140788713*n^3 + 210641238*n^2 - 166531044*n + 54330068)*a(n) = (n-1)^2*(109683470*n^10 - 1564397770*n^9 + 9692963299*n^8 - 34227043418*n^7 + 75994068609*n^6 - 110509975758*n^5 + 106422212572*n^4 - 67092633284*n^3 + 26619112256*n^2 - 6034674112*n + 596279040)*a(n-1) - (1736373702*n^12 - 31711114890*n^11 + 261518988565*n^10 - 1286766506127*n^9 + 4203065855621*n^8 - 9590033857111*n^7 + 15649936441072*n^6 - 18370855225904*n^5 + 15360506258964*n^4 - 8896962441876*n^3 + 3377234408016*n^2 - 751582555104*n + 73915071552)*a(n-2) - (n-2)^2*(36279917*n^10 - 590014481*n^9 + 4216923435*n^8 - 17398379754*n^7 + 45760527058*n^6 - 79915647314*n^5 + 93501944898*n^4 - 72055169071*n^3 + 34824212688*n^2 - 9481092472*n + 1100757336)*a(n-3) - (n-2)^2*(843719*n^6 - 5284077*n^5 + 13396609*n^4 - 17487127*n^3 + 12303648*n^2 - 4393232*n + 621656)*(n-3)^4*a(n-4). - _Vaclav Kotesovec_, Nov 04 2012
%t Table[Sum[Binomial[n,k]^2*Binomial[n+k,k]^3,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Nov 04 2012 *)
%o (PARI) a(n) = sum(k=0, n, binomial(n, k)^2*binomial(n+k,k)^3); \\ _Michel Marcus_, Mar 10 2016
%Y Cf. A218693, A112019, A014178, A014180, A218689, A218692.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 28 2005
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