A012474 - OEIS

%I #7 Jul 18 2021 12:12:31

%S 1,-3,-15,357,22305,-582483,-217683375,-2001865323,7258418490945,

%T 826442897487837,-580853295560646735,-204631967729205027003,

%U 85920364932388858911585,79053144190920049745846157

%N Expansion of arcsin(cos(x)*sin(x)) = x - 3/3!*x^3 - 15/5!*x^5 + 357/7!*x^7 + 22305/9!*x^9...

%F E.g.f.: arcsin(cos(x)*sin(x)) = sum a(n)x^(2n+1)/(2n+1)!.

%t With[{nn=30},Take[CoefficientList[Series[ArcSin[Cos[x]Sin[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Jul 18 2021 *)

%o (PARI) a(n)=local(sx); if(n<0, 0, sx=sin(x+O(x^(2*n+2))); (2*n+1)!*polcoeff(subst(serreverse(sx), x, subst(sx, x, x*2)/2), 2*n+1))

%K sign

%O 0,2

%A Patrick Demichel (patrick.demichel(AT)hp.com)