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Expansion of e.g.f. cos(log(1+tan(x))).
1

%I #24 Sep 08 2022 08:44:37

%S 1,0,-1,3,-18,100,-726,5698,-52028,522720,-5849876,71473468,

%T -951628488,13674233040,-211091632376,3478906531688,-60920066577648,

%U 1127785660994560,-21969631512312976,448167387287730608

%N Expansion of e.g.f. cos(log(1+tan(x))).

%H G. C. Greubel, <a href="/A009021/b009021.txt">Table of n, a(n) for n = 0..250</a>

%t With[{nn=25},CoefficientList[Series[Cos[Log[1+Tan[x]]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 26 2011 *)

%o (Maxima)

%o a(n):=if n=0 them 1 else sum(sum((stirling1(n-2*i,2*m)*sum(binomial(j-1,n-2*i-1)*j!*2^(n-j)*(-1)^(n-i+j+m)*stirling2(n,j),j,n-2*i,n))/(n-2*i)!,i,0,n/2-m),m,0,n/2); /* _Vladimir Kruchinin_, Jun 20 2011 */

%o (PARI) x='x+O('x^30); Vec(serlaplace(cos(log(1+tan(x))))) \\ _G. C. Greubel_, Jul 22 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cos(Log(1+Tan(x))))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 22 2018

%K sign,easy

%O 0,4

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997