A189772 - OEIS

%I #17 Jan 15 2018 14:44:05

%S 1,1,0,-1,2,0,-20,100,-180,-1540,18800,-99900,-35000,6978400,

%T -81010800,418106000,2652650000,-89962470000,1078639120000,

%U -4572607130000,-102140361180000,2809462217120000,-34739768494600000,80549366231880000,7017075372032440000

%N The n-th derivative of exp(2*arctan(x) - Pi/2), evaluated at x=1.

%C The n-th derivative of exp(2*arctan(x)) A(n,x) = (n!*e^(2*arctan(x))*sum(m=1..n, sum(k=m..n, (sum(j=m..k, (2^j*Stirling1(j,m)*binomial(k-1,j-1))/j!))*((-1)^((m+3*k)/2)+(-1)^((k-m)/2))*(-1)^(n-k)*binomial(n-1,k-1)*x^(n-k))))/(2*(x^2+1)^n).

%H Alois P. Heinz, <a href="/A189772/b189772.txt">Table of n, a(n) for n = 0..477</a>

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1104.5065">Derivation of Bell Polynomials of the Second Kind</a>, arXiv:1104.5065

%F a(n) = n!*sum(m=1..n, 2^(-n-1)*sum(k=m..n, (sum(j=m..k, (2^j*Stirling1(j,m) * binomial(k-1,j-1))/j!))*((-1)^((m+3*k)/2) + (-1)^((k-m)/2))*(-1)^(n-k)*binomial(n-1,k-1))), n>0, a(0)=1.

%F E.g.f.: exp(2*arctan(x+1)-Pi/2). - _Alois P. Heinz_, Sep 27 2016

%t f[x_] := Exp[2*ArcTan[x] - Pi/2];  a[n_] := Derivative[n][f][1]; Table[a[n], {n, 0, 20}] (* or *) a[n_] := n!* Sum[ 2^(-n-1)*Sum[ (Sum[ (2^j*StirlingS1[j, m]*Binomial[k-1, j-1])/j!, {j, m, k}])*((-1)^((m + 3*k)/2) + (-1)^((k - m)/2))*(-1)^(n-k)*Binomial[n-1, k-1], {k, m, n}], {m, 1, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 22 2013 *)

%o (Maxima)

%o a(n):=n!*sum(2^(-n-1)*sum((sum((2^j*stirling1(j,m)*binomial(k-1,j-1))/j!,j,m,k))*((-1)^((m+3*k)/2)+(-1)^((k-m)/2))*(-1)^(n-k)*binomial(n-1,k-1),k,m,n),m,1,n);

%K sign

%O 0,5

%A _Vladimir Kruchinin_, Apr 27 2011