A096719 Numerators of terms in series expansion of arctan(arcsin(x)).

1, -1, 13, -173, 12409, -123379, 29518679, -889424791, 92273231203, 3836321172631, 22487012578592981, 2865860401219263691, 35970731592390474409, 277817773865257308429491, 1687365015862907602230599, 22415401434548677685890690591, 5789220720660809183499012532793, 2838956049184596030388390046497291

Offset: 0

Author

N. J. A. Sloane, Aug 15 2004

Keywords

• sign
• frac

Examples

			arctan(arcsin(x)) = x - 1/6*x^3 + 13/120*x^5 - 173/5040*x^7 + 12409/362880*x^9 - 123379/13305600*x^11 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A096720, A096718, A096664, A096671, A096712, A096716, A045688, A045689.

Programs

Mathematica

Numerator[Take[CoefficientList[Series[ArcTan[ArcSin[x]], {x,0,40}], x], {2, -1, 2}]] (* G. C. Greubel, Nov 18 2016 *)

Maxima

a(n):=b(2*n+1);
b(n):=num(1/n*sum((1-(-1)^(m))*(-1)^((m-1)/2)*(1+(-1)^(n-m))/4*sum((sum(binomial(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i-j)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!,i,0,floor(j/2)),j,1,k))*binomial(k+n-1,n-1),k,1,n-m),m,1,n-1)+(1-(-1)^(n))/(2)*(-1)^((n-1)/2)/n); /* Vladimir Kruchinin, May 02 2011 */

Formula

a(n) = b(2*n+1), b(n) = numerator(1/n*sum(m=1..n-1, (1-(-1)^(m))*(-1)^((m-1)/2)*(1+(-1)^(n-m))/4*sum(k=1..n-m, (sum(j=1..k, binomial(k,j)*2^(1-j)* sum(i=0..floor(j/2), (-1)^((n-m)/2-i-j)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!)))*binomial(k+n-1,n-1)))+(1-(-1)^(n))/(2)*(-1)^((n-1)/2)/n). - Vladimir Kruchinin, May 02 2011