%I #50 Jan 08 2023 02:40:17
%S 1,12,1680,665280,518918400,670442572800,1295295050649600,
%T 3497296636753920000,12576278705767096320000,
%U 58102407620643984998400000,335367096786357081410764800000,2365008766537390138108713369600000,20007974164906320568399715106816000000
%N a(n) = (4n)!/(2n)!.
%C Absolute value of the coefficients in the expansion of cos(x^2). - clarified by _Muniru A Asiru_, Jul 26 2018
%C Bisection of sequence A001813. - _Gary W. Adamson_, Jul 19 2011
%C Expansion of cosh(x^2) in powers of x^4. - _G. C. Greubel_, Jul 26 2018
%H Vincenzo Librandi, <a href="/A009120/b009120.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = 4^n * A101485(n).
%F Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int((1/4)*exp(-1/4*sqrt(x))/(sqrt(Pi)*x^(3/4)), x=0..infinity), n=0, 1... - _Karol A. Penson_, Sep 19 2001
%F From _Gary W. Adamson_, Jul 19 2011: (Start)
%F a(n) = upper left term of M^(2n), where M = an infinite square production matrix as follows:
%F 2, 2, 0, 0, 0, 0, ...
%F 4, 4, 4, 0, 0, 0, ...
%F 6, 6, 6, 6, 0, 0, ...
%F 8, 8, 8, 8, 8, 0, ...
%F ... (End)
%F Sum_{n>=0} 1/a(n) = 1 + (1/4) * exp(1/4) * sqrt(Pi) * erf(1/2) - (1/4) * exp(-1/4) * sqrt(Pi) * erfi(1/2), where erf is the error function and erfi is the imaginary error function. - _Amiram Eldar_, Jan 08 2023
%p seq(coeff(series(factorial(n)*cosh(x^2), x,n+1),x,n),n=0..50,4); # _Muniru A Asiru_, Jul 27 2018
%t Table[(4n)!/(2n)!,{n,0,10}] (* or *) With[{nn=60},Abs[Take[ CoefficientList[ Series[ Cos[x^2],{x,0,nn}],x] Range[0,nn]!,{1,-1,4}]]] (* _Harvey P. Dale_, Mar 27 2012 *)
%o (Magma) [Factorial(4*n)/Factorial(2*n): n in [0..15]]; // _Vincenzo Librandi_, Jul 20 2011
%o (PARI) for(n=0, 20, print1((4*n)!/(2*n)!, ", ")) \\ _G. C. Greubel_, Jul 26 2018
%o (PARI) x='x+O('x^120); v=Vec(serlaplace(cosh(x^2))); vector(#v\4, n, v[4*n-3]) \\ _G. C. Greubel_, Jul 26 2018
%o (GAP) List([0..25],n->Factorial(4*n)/Factorial(2*n)); # _Muniru A Asiru_, Jul 26 2018
%Y Cf. A001813, A101485.
%K nonn,easy
%O 0,2
%A _R. H. Hardin_
%E Extended by _Olivier GĂ©rard_, Mar 01 1997