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A096719 Numerators of terms in series expansion of arctan(arcsin(x)). 8

%I

%S 1,-1,13,-173,12409,-123379,29518679,-889424791,92273231203,

%T 3836321172631,22487012578592981,2865860401219263691,

%U 35970731592390474409,277817773865257308429491,1687365015862907602230599,22415401434548677685890690591,5789220720660809183499012532793,2838956049184596030388390046497291

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

%H G. C. Greubel, <a href="/A096719/b096719.txt">Table of n, a(n) for n = 0..200</a>

%F 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

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

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

%o (Maxima)

%o a(n):=b(2*n+1);

%o 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 */

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

%K sign,frac

%O 0,3

%A _N. J. A. Sloane_, Aug 15 2004

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Last modified June 1 02:09 EDT 2020. Contains 334758 sequences. (Running on oeis4.)