A009564 E.g.f. sin(x^2)/2, coefficients of x^(4*n + 2).

1, -60, 15120, -8648640, 8821612800, -14079294028800, 32382376266240000, -101421602465863680000, 415017197290314178560000, -2149789081963827444940800000, 13750050968240640337841356800000, -106425394494182556214892101632000000, 980390734080409707851586040233984000000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..182

Formula

a(n) = (-1)^n*(2+4*n)!/(2*(1+2*n)!) = (-1)^n*A001813(2*n+1)/2. - Robert Israel, Dec 21 2015

From Amiram Eldar, Sep 02 2025: (Start)

a(n) = A024343(n)/2.

Sum_{n>=0} 1/a(n) = sqrt(2*Pi) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.

Sum_{n>=0} (-1)^n/a(n) = (sqrt(Pi)/2) * (exp(1/4) * erf(1/2) + erfi(1/2) / exp(1/4)). (End)

Maple

seq(i!*coeff(series(sin(x^2)/2, x, 4*i+4), x, i), i=2..54, 4); # Peter Luschny, Dec 14 2012

Mathematica

nmax = 12; coes = CoefficientList[ Series[ Sin[x^2]/2, {x, 0, 4*nmax + 2}], x]; a[n_] := coes[[4*n + 3]]*(4*n + 2)!; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 14 2012 *)
Table[(-1)^n (2 + 4 n)!/(2 (1 + 2 n)!), {n, 0, 25}] (* Vincenzo Librandi, Dec 22 2015 *)

Prog

(SageMath)
def A009564(n):
    return falling_factorial(4*n+2, 2*n+1)/(2*(-1)^n)
[A009564(n) for n in (0..12)]  # Peter Luschny, Dec 14 2012
(Magma) [(-1)^n*Factorial(2+4*n)/(2*Factorial(1+2*n)): n in [0..20]]; // Vincenzo Librandi, Dec 22 2015
(PARI) a(n) = (-1)^n*(2+4*n)!/(2*(1+2*n)!); \\ Altug Alkan, Dec 22 2015

Crossrefs

Cf. A001813, A103639, A024343, A075069.

Sequence in context: A184890 A295598 A113424 * A383929 A269762 A291912

Adjacent sequences: A009561 A009562 A009563 * A009565 A009566 A009567

Keyword

sign

Author

R. H. Hardin

Extensions

Extended with signs Mar 1997

Definition corrected and terms a(10)-a(12) from Peter Luschny, Dec 14 2012

Status

approved