A024343 Expansion of e.g.f. sin(x^2) in powers of x^(4*n + 2).

2, -120, 30240, -17297280, 17643225600, -28158588057600, 64764752532480000, -202843204931727360000, 830034394580628357120000, -4299578163927654889881600000, 27500101936481280675682713600000

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• changed

Comments

Absolute values are coefficients of expansion of sinh(x^2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..175

Crossrefs

Bisection of A001813.

Cf. A009564.

Programs

Magma

F:=Factorial;; [(-1)^n*F(4*n+2)/F(2*n+1) : n in [0..30]]; // G. C. Greubel, Jan 29 2022

Mathematica

Table[(-1)^n*(2*n+1)!*Binomial[4*n+2, 2*n+1], {n,0,30}] (* G. C. Greubel, Jan 29 2022 *)

PARI

a(n)=polcoeff(serlaplace(sin(x^2)),4*n+2)

PARI

a(n)=(-1)^n*(4*n+2)!/(2*n+1)!

Sage

f=factorial; [(-1)^n*f(4*n+2)/f(2*n+1) for n in (0..30)] # G. C. Greubel, Jan 29 2022

Formula

a(n) = (-1)^n * (4*n+2)! / (2*n+1)!.

E.g.f.: [x^(4*n+2)] sin(x^2)

a(n) = 2 * A009564(n). - Sean A. Irvine, Jul 01 2019

From Amiram Eldar, Sep 02 2025: (Start)

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

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Edited by Ralf Stephan, Mar 25 2004

Name edited by Michel Marcus, Jul 01 2019