A110708 E.g.f. log(1+arctan(x)).

0, 1, -1, 0, 2, 8, -64, -112, 2064, 8192, -157056, -599808, 16072704, 80010240, -2484268032, -13537247232, 506459129856, 3160676007936, -135526008225792, -929451393220608, 45507663438741504, 343173318976733184, -18834884514478817280, -154043745649772986368

Offset: 0

Author

Vladimir Kruchinin, Jun 12 2011

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..450

Programs

Mathematica

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

Maxima

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);

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;
cons(0,makelist((n-1)!*b[n],n,1,100)); /* Tani Akinari, Oct 30 2017 */

PARI

my(x='x+O('x^50)); concat([0], Vec(serlaplace(log(1 + atan(x))))) \\ G. C. Greubel, Sep 06 2017

Formula

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).