A101485 a(n) = (4n)! / ( 4^n * (2n)! ).
1, 3, 105, 10395, 2027025, 654729075, 316234143225, 213458046676875, 191898783962510625, 221643095476699771875, 319830986772877770815625, 563862029680583509947946875, 1192568192774434123539907640625, 2980227913743310874726229193921875
Offset: 0
Programs
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Maple
seq(4^n*pochhammer(1/2,2*n), n=0..12); # Peter Luschny, Jul 05 2011
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Mathematica
f[n_] := 4^n*Pochhammer[1/2, 2 n]; Array[f, 13, 0] (* Robert G. Wilson v, Jul 05 2011 *)
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PARI
a(n)=(4*n)!/(2*n)!>>(2*n) \\ Charles R Greathouse IV, Jul 06 2011
Formula
sin(arcsin(2x)/2) = x + 3x^3/3! + 105x^5/5! + 10395x^7/7! + ...
E.g.f.: cosh(x^2/2). - Paul Barry, Sep 28 2010
a(n) = 4^n*Gamma(2*n+1/2) / Gamma(1/2). - Peter Luschny, Jul 05 2011
Hypergeom. recurrence: a(n) -(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Sep 21 2012
Sum_{n>=0} 1/a(n) = 1 + (1/2) * sqrt(e*Pi/2) * erf(1/sqrt(2)) - (1/2) * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)), where erf is the error function and erfi is the imaginary error function. - Amiram Eldar, Jan 08 2023