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%I #16 Dec 29 2023 12:57:48
%S 0,1,18,400,9800,254016,6830208,188457984,5300380800,151289881600,
%T 4369251780608,127394382495744,3743979352236032,110768619888640000,
%U 3295931587706880000,98555678764852838400,2959750227906986803200
%N a(n) = 2^(n-1)*binomial(2n-1,n-1)^2.
%H Vincenzo Librandi, <a href="/A116421/b116421.txt">Table of n, a(n) for n = 0..100</a>
%F G.f.: 1+(K(32x)-1)/4 where K(k)=Elliptic_F(pi/2,k) is the complete Elliptic integral of the first kind;
%F e.g.f.: BesselI(0, 2*sqrt(2)x)*BesselI(1, 2*sqrt(2)x)/sqrt(2);
%F a(n) = 2^(n+1)*(binomial(2n,n)/4)^2 - 0^n/8.
%F Conjecture: n^2*a(n) - (2*n-1)^2*a(n-1) = 0. - _R. J. Mathar_, Nov 16 2011
%t Join[{0},Table[2^(n-1) Binomial[2n-1,n-1]^2,{n,20}]] (* _Harvey P. Dale_, Dec 29 2023 *)
%o (Magma) [2^(n-1)*Binomial(2*n-1,n-1)^2: n in [0..20]]; // _Vincenzo Librandi_, Nov 17 2011
%o (PARI) a(n)=binomial(2*n-1,n-1)^2<<(n-1) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y Cf. A060150.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Feb 14 2006
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