%I #21 Jan 30 2022 09:50:49

%S 2,-120,30240,-17297280,17643225600,-28158588057600,64764752532480000,

%T -202843204931727360000,830034394580628357120000,

%U -4299578163927654889881600000,27500101936481280675682713600000

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

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

%H G. C. Greubel, <a href="/A024343/b024343.txt">Table of n, a(n) for n = 0..175</a>

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

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

%F a(n) = 2 * A009564(n). - _Sean A. Irvine_, Jul 01 2019

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

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

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

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

%o (Magma) F:=Factorial;; [(-1)^n*F(4*n+2)/F(2*n+1) : n in [0..30]]; // _G. C. Greubel_, Jan 29 2022

%Y Bisection of A001813.

%Y Cf. A009564.

%K sign

%O 0,1

%A _R. H. Hardin_

%E Extended with signs by _Olivier Gérard_, Mar 15 1997

%E Edited by _Ralf Stephan_, Mar 25 2004

%E Name edited by _Michel Marcus_, Jul 01 2019