A191512 Arctan(x*cos(x)) = Sum_{n >= 0} a(n)*x^(2n+1)/(2n+1)!.
1, -5, 89, -4717, 449073, -69090581, 15583801609, -4846181282685, 1987373846425697, -1039121484066627877, 674707915373741222841, -532627526452975709882765, 502375568363623615781076625, -557965947638266639781208500277, 720767702359064719935712626879593, -1071470941893105999704454019614019741
Offset: 0
Keywords
Examples
x-(5/6)*x^3+(89/120)*x^5-(4717/5040)*x^7+(49897/40320)*x^9-(9870083/5702400)*x^11+(15583801609/6227020800)*x^13-(35897639131/9686476800)*x^15+...
Programs
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Mathematica
With[{nn=40},Take[CoefficientList[Series[ArcTan[x Cos[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Jul 26 2020 *) -
Maxima
a(n):=(-1)^(n-1)*(2*n-1)!*sum(((sum((2*m-1-2*i)^(2*n-2*m)*binomial(2*m-1,i),i,0,(m-1))))/((2*m-1)*2^(2*m-2)*(2*n-2*m)!),m,1,n);
Formula
a(n)=(-1)^(n-1)*(2*n-1)!*sum(m=1..n, ((sum(i=0..(m-1),(2*m-1-2*i)^(2*n-2*m)*binomial(2*m-1,i))))/((2*m-1)*2^(2*m-2)*(2*n-2*m)!))for n>=1, a(0)=0.
Extensions
Edited by N. J. A. Sloane, Jul 25 2020 at the suggestion of Harvey P. Dale.