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A009120 a(n) = (4n)!/(2n)!. 2

%I

%S 1,12,1680,665280,518918400,670442572800,1295295050649600,

%T 3497296636753920000,12576278705767096320000,

%U 58102407620643984998400000,335367096786357081410764800000

%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)

%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

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Last modified September 20 07:21 EDT 2020. Contains 337264 sequences. (Running on oeis4.)