login
E.g.f. arctan(log(1+tanh(x))).
0

%I #10 Jan 15 2018 14:43:09

%S 0,1,-1,-2,14,-6,-406,2220,13768,-280224,501824,34875600,-394160416,

%T -3296452368,129418793776,-532969950240,-35512135858048,

%U 654965382945408,5943454902108032,-395494767434053632,2913398252129581312

%N E.g.f. arctan(log(1+tanh(x))).

%F a(n)=sum(m=0..(n-1)/2, ((-1)^(m)*(2*m+1)!*sum(r=2*m+1..n, (stirling1(r,2*m+1)*sum(k=r..n, binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k)))/r!))/(2*m+1)), n>0.

%t With[{nn=30},CoefficientList[Series[ArcTan[Log[1+Tanh[x]]],{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Jan 03 2013 *)

%o (Maxima)

%o a(n):=sum(((-1)^(m)*(2*m+1)!*sum((stirling1(r,2*m+1)*sum(binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k),k,r,n))/r!,r,2*m+1,n))/(2*m+1),m,0,(n-1)/2);

%K sign

%O 0,4

%A _Vladimir Kruchinin_, Jun 21 2011