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

1, 12, 1680, 665280, 518918400, 670442572800, 1295295050649600, 3497296636753920000, 12576278705767096320000, 58102407620643984998400000, 335367096786357081410764800000, 2365008766537390138108713369600000, 20007974164906320568399715106816000000

Offset

0,2

Comments

Absolute value of the coefficients in the expansion of cos(x^2). - clarified by Muniru A Asiru, Jul 26 2018

Bisection of sequence A001813. - Gary W. Adamson, Jul 19 2011

Expansion of cosh(x^2) in powers of x^4. - G. C. Greubel, Jul 26 2018

Links

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

Formula

a(n) = 4^n * A101485(n).

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

From Gary W. Adamson, Jul 19 2011: (Start)

a(n) = upper left term of M^(2n), where M = an infinite square production matrix as follows:

2, 2, 0, 0, 0, 0, ...

4, 4, 4, 0, 0, 0, ...

6, 6, 6, 6, 0, 0, ...

8, 8, 8, 8, 8, 0, ...

... (End)

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

Maple

seq(coeff(series(factorial(n)*cosh(x^2), x, n+1), x, n), n=0..50, 4); # Muniru A Asiru, Jul 27 2018

Mathematica

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 *)

Prog

(Magma) [Factorial(4*n)/Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Jul 20 2011
(PARI) for(n=0, 20, print1((4*n)!/(2*n)!, ", ")) \\ G. C. Greubel, Jul 26 2018
(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
(GAP) List([0..25], n->Factorial(4*n)/Factorial(2*n)); # Muniru A Asiru, Jul 26 2018

Crossrefs

Cf. A001813, A101485.

Sequence in context: A271436 A034280 A146201 * A078928 A202968 A013717

Adjacent sequences: A009117 A009118 A009119 * A009121 A009122 A009123

Keyword

nonn,easy

Author

R. H. Hardin

Extensions

Extended by Olivier Gérard, Mar 01 1997

Status

approved