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Self-convolution of Sum(binomial(2*n, i), i=0..n).
2

%I #18 Apr 17 2014 06:53:40

%S 1,6,31,150,699,3178,14198,62604,273235,1182786,5085666,21743956,

%T 92522206,392066340,1655432524,6967724312,29245179267,122442487474,

%U 511487386730,2132341655556,8873167793578,36861311739308,152895342950196,633290273209000,2619653638855214,10823294835350388

%N Self-convolution of Sum(binomial(2*n, i), i=0..n).

%H Fung Lam, <a href="/A240879/b240879.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. = (g.f. of A032443)^2.

%F n*a(n) = 32*(2*n-3)*a(n-3) + 48*(1-n)*a(n-2) + 6*(2*n-1)*a(n-1).

%F Asymptotics: a(n) ~ 2^(2*n)*((n+2)/4 + sqrt(n/Pi)).

%F Recurrence: (n-2)*n*a(n) = 2*n*(4*n-7)*a(n-1) - 8*(n-1)*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Apr 16 2014

%t CoefficientList[Series[((1/Sqrt[1-4*x] + 1/(1-4*x))/2)^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 16 2014 *)

%Y Cf. A032443.

%K nonn

%O 0,2

%A _Fung Lam_, Apr 13 2014