This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A009120 #54 May 04 2025 01:54:18
%S A009120 1,12,1680,665280,518918400,670442572800,1295295050649600,
%T A009120 3497296636753920000,12576278705767096320000,
%U A009120 58102407620643984998400000,335367096786357081410764800000,2365008766537390138108713369600000,20007974164906320568399715106816000000
%N A009120 a(n) = (4*n)!/(2*n)!.
%C A009120 Absolute value of the coefficients in the expansion of cos(x^2). - clarified by _Muniru A Asiru_, Jul 26 2018
%C A009120 Bisection of sequence A001813. - _Gary W. Adamson_, Jul 19 2011
%C A009120 Expansion of cosh(x^2) in powers of x^4. - _G. C. Greubel_, Jul 26 2018
%H A009120 Vincenzo Librandi, <a href="/A009120/b009120.txt">Table of n, a(n) for n = 0..100</a>
%F A009120 a(n) = 4^n * A101485(n).
%F A009120 Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*(1/4)*exp(-1/4*sqrt(x))/(sqrt(Pi)*x^(3/4)) dx, n >= 0. - _Karol A. Penson_, Sep 19 2001
%F A009120 From _Gary W. Adamson_, Jul 19 2011: (Start)
%F A009120 a(n) = upper left term of M^(2n), where M = an infinite square production matrix as follows:
%F A009120 2, 2, 0, 0, 0, 0, ...
%F A009120 4, 4, 4, 0, 0, 0, ...
%F A009120 6, 6, 6, 6, 0, 0, ...
%F A009120 8, 8, 8, 8, 8, 0, ...
%F A009120 ... (End)
%F A009120 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 A009120 seq(coeff(series(factorial(n)*cosh(x^2), x,n+1),x,n),n=0..50,4); # _Muniru A Asiru_, Jul 27 2018
%t A009120 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 A009120 (Magma) [Factorial(4*n)/Factorial(2*n): n in [0..15]]; // _Vincenzo Librandi_, Jul 20 2011
%o A009120 (PARI) for(n=0, 20, print1((4*n)!/(2*n)!, ", ")) \\ _G. C. Greubel_, Jul 26 2018
%o A009120 (PARI) my(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 A009120 (GAP) List([0..25],n->Factorial(4*n)/Factorial(2*n)); # _Muniru A Asiru_, Jul 26 2018
%Y A009120 Cf. A001813, A101485.
%K A009120 nonn,easy
%O A009120 0,2
%A A009120 _R. H. Hardin_
%E A009120 Extended by _Olivier Gérard_, Mar 01 1997