A191500 - OEIS

%I #7 Dec 20 2021 14:01:44

%S 1,1,-1,-5,17,249,-1489,-27453,237537,6037041,-68649441,-2107439157,

%T 29789919345,1092524775081,-18492402857265,-781266357571053,

%U 15425010795541185,739391174869277025,-16695627953904545985,-893468264544135234405,22725280358984904476625

%N E.g.f. sqrt(1+arctan(2*x))

%F a(n)=2*n!*sum(k=0..(n-1)/2, ((-1)^(n+k+1)*binomial(2*n-4*k-2,n-2*k-1)*(n-2*k-1)!*(sum(i=0..2*k, (2^(i+4*k-n)*stirling1(i+n-2*k,n-2*k)*binomial(n-1,i+n-2*k-1))/(i+n-2*k)!)))), n>0, a(0)=1.

%t With[{nn=20},CoefficientList[Series[Sqrt[1+ArcTan[2x]],{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Dec 20 2021 *)

%o (Maxima)

%o a(n):=2*n!*sum(((-1)^(n+k+1)*binomial(2*n-4*k-2,n-2*k-1)*(n-2*k-1)!*(sum((2^(i+4*k-n)*stirling1(i+n-2*k,n-2*k)*binomial(n-1,i+n-2*k-1))/(i+n-2*k)!,i,0,2*k))),k,0,(n-1)/2);

%K sign

%O 0,4

%A _Vladimir Kruchinin_, Jun 03 2011