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%I #11 Feb 07 2015 16:58:07
%S 1,-1,5,-109,4521,-330681,36468717,-5721101541,1205289090513,
%T -328897528901361,112769846051251797,-47472161654530776285,
%U 24068680135210975714425,-14466929567049590650593705
%N cos(arcsin(arctan(x)))=1-1/2!*x^2+5/4!*x^4-109/6!*x^6+4521/8!*x^8...
%F a(n) = (2*n)!*sum(k=1..2*n, ((2*k)!*binomial(2*k-2,k-1)*(-1)^(n+k+1)*sum(i=2*k..2*n, (2^(i+1)*stirling1(i,2*k)*binomial(2*n-1,i-1))/i!))/(k*2^(4*k))) with n>0, a(0)=1. [_Vladimir Kruchinin_, Oct 08 2012]
%t With[{nn=30},Take[CoefficientList[Series[Cos[ArcSin[ArcTan[x]]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Feb 07 2015 *)
%o (Maxima) a[n]:=if n=0 then 1 else (2*n)!*sum(((2*k)!*binomial(2*k-2,k-1)*(-1)^(n+k+1)*sum((2^(i+1)*stirling1(i,2*k)*binomial(2*n-1,i-1))/i!, i, 2*k, 2*n))/(k*2^(4*k)), k, 1, 2*n); makelist(a[n], n, 0, 13); [_Vladimir Kruchinin_, Oct 08 2012]
%Y Cf. A012254.
%K sign
%O 0,3
%A Patrick Demichel (patrick.demichel(AT)hp.com)
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