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Alternate partial sums of binomial(2n,n)^2.
3

%I #22 Nov 02 2016 13:44:16

%S 1,3,33,367,4533,58971,794805,10983819,154653081,2209251319,

%T 31925528217,465708778407,6846750893929,101325729466071,

%U 1508015866093929,22553429144856471,338744206097695629,5106973783924992771,77251106929381097229,1172036566162209342771

%N Alternate partial sums of binomial(2n,n)^2.

%H Seiichi Manyama, <a href="/A228002/b228002.txt">Table of n, a(n) for n = 0..833</a>

%F Recurrence: n^2*a(n) = (3*n-2)*(5*n-2)*a(n-1) + 4*(2*n-1)^2*a(n-2).

%F a(n) ~ 16^(n+1)/(17*Pi*n).

%p series(2*EllipticK(4*x^(1/2))/(Pi*(1+x)),x=0,20)

%t Table[Sum[(-1)^(n-k)*Binomial[2*k,k]^2, {k, 0, n}], {n, 0, 20}]

%Y Cf. A115257, A188679, A188680.

%K nonn,easy

%O 0,2

%A _Vaclav Kotesovec_, Aug 07 2013