A189772 The n-th derivative of exp(2*arctan(x) - Pi/2), evaluated at x=1.

1, 1, 0, -1, 2, 0, -20, 100, -180, -1540, 18800, -99900, -35000, 6978400, -81010800, 418106000, 2652650000, -89962470000, 1078639120000, -4572607130000, -102140361180000, 2809462217120000, -34739768494600000, 80549366231880000, 7017075372032440000

Offset

0,5

Comments

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..477

Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065

Formula

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.

E.g.f.: exp(2*arctan(x+1)-Pi/2). - Alois P. Heinz, Sep 27 2016

Mathematica

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 *)

Prog

(Maxima)
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);

Crossrefs

Sequence in context: A365862 A209868 A182661 * A217311 A266167 A290465

Adjacent sequences: A189769 A189770 A189771 * A189773 A189774 A189775

Keyword

sign

Author

Vladimir Kruchinin, Apr 27 2011

Status

approved