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Sum_{k=0..n} k*binomial(n,k)^2*binomial(2*k,k).
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%I #18 Apr 12 2014 02:08:08

%S 0,2,20,186,1704,15510,140676,1273230,11508048,103919022,937787100,

%T 8458728630,76269112200,687496910490,6195793616460,55827244680930,

%U 502959206683296,4530723835554270,40809306881317068,367548287590324902,3310080578306654520

%N Sum_{k=0..n} k*binomial(n,k)^2*binomial(2*k,k).

%H Vincenzo Librandi, <a href="/A226312/b226312.txt">Table of n, a(n) for n = 0..200</a>

%H Bruno Haible, <a href="http://www.haible.de/bruno/papers/math/combinatorics/combident/">Combinatorial proof of a binomial identity</a>, 1992.

%F a(n) ~ 3^(2*n+1/2)/(2*Pi). - _Vaclav Kotesovec_, Jun 10 2013

%F Recurrence: (n-2)*n*(n-1)*a(n) = (n-2)*(10*n^2-10*n+3)*a(n-1) - 9*(n-1)^3*a(n-2). - _Vaclav Kotesovec_, Jun 10 2013

%F G.f.: 2*x*((5+3*x)*(1-9*x)^2*hypergeom([2/3, 2/3],[1],-27*x*(1-x)^2/(1-9*x)^2)-4*(1-x)*(1+3*x)^3*hypergeom([5/3, 5/3],[2],-27*x*(1-x)^2/(1-9*x)^2))/(1-9*x)^(13/3). - _Mark van Hoeij_, Apr 11 2014

%p f:=n->add(k*binomial(n,k)^2*binomial(2*k,k),k=0..n);

%p [seq(f(n),n=0..40)];

%t Table[Sum[k*Binomial[n,k]^2*Binomial[2*k,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 10 2013 *)

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jun 08 2013