A110708 - OEIS

%I #20 Oct 29 2017 21:47:00

%S 0,1,-1,0,2,8,-64,-112,2064,8192,-157056,-599808,16072704,80010240,

%T -2484268032,-13537247232,506459129856,3160676007936,-135526008225792,

%U -929451393220608,45507663438741504

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

%H G. C. Greubel, <a href="/A110708/b110708.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = n!*Sum_{m=0..(n-1)/2} (2^(2*m-n)*(n-2*m)!*(-1)^(n-m-1) * Sum_{i=0..2*m} (2^(i+n-2*m)*Stirling1(n-2*m+i,n-2*m)*binomial(n-1,n-2*m+i-1))/(n-2*m+i)!))/(n-2*m).

%t With[{nn = 50}, CoefficientList[Series[Log[1 + ArcTan[x]], {x, 0, nn}], x]*Range[0, nn]!] (* _G. C. Greubel_, Sep 06 2017 *)

%o (Maxima)

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

%o (Maxima) b[1]:1$ b[n]:=sum((-1)^(k+1)*b[n-1-2*k]/(2*k+1),k,0,floor(n/2)-1)+((%i)^(n-1)+(-%i)^(n-1))/2;

%o cons(0,makelist((n-1)!*b[n],n,1,100)); /* _Tani Akinari_, Oct 30 2017 */

%o (PARI) x='x+O('x^50); concat([0], Vec(serlaplace(log(1 + atan(x))))) \\ _G. C. Greubel_, Sep 06 2017

%K sign

%O 0,5

%A _Vladimir Kruchinin_, Jun 12 2011