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

0,2

Links

Table of n, a(n) for n=0..15.

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.

Example

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+...

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 *)

Prog

(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);

Crossrefs

Sequence in context: A339001 A330605 A028353 * A015085 A258181 A355082

Adjacent sequences: A191509 A191510 A191511 * A191513 A191514 A191515

Keyword

sign

Author

Vladimir Kruchinin, Jun 13 2011

Extensions

Edited by N. J. A. Sloane, Jul 25 2020 at the suggestion of Harvey P. Dale.

Status

approved