A116421 a(n) = 2^(n-1)*binomial(2n-1,n-1)^2.

0, 1, 18, 400, 9800, 254016, 6830208, 188457984, 5300380800, 151289881600, 4369251780608, 127394382495744, 3743979352236032, 110768619888640000, 3295931587706880000, 98555678764852838400, 2959750227906986803200

Offset: 0

Author

Paul Barry, Feb 14 2006

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A060150.

Programs

Magma

[2^(n-1)*Binomial(2*n-1,n-1)^2: n in [0..20]]; // Vincenzo Librandi, Nov 17 2011

Mathematica

Join[{0},Table[2^(n-1) Binomial[2n-1,n-1]^2,{n,20}]] (* Harvey P. Dale, Dec 29 2023 *)

PARI

a(n)=binomial(2*n-1,n-1)^2<<(n-1) \\ Charles R Greathouse IV, Oct 23 2023

Formula

G.f.: 1+(K(32x)-1)/4 where K(k)=Elliptic_F(pi/2,k) is the complete Elliptic integral of the first kind;

e.g.f.: BesselI(0, 2*sqrt(2)x)*BesselI(1, 2*sqrt(2)x)/sqrt(2);

a(n) = 2^(n+1)*(binomial(2n,n)/4)^2 - 0^n/8.

Conjecture: n^2*a(n) - (2*n-1)^2*a(n-1) = 0. - R. J. Mathar, Nov 16 2011