A177146 n-th derivative of arctan(x) at x = 1, n >= 4.

0, -3, 15, -45, 0, 1260, -11340, 56700, 0, -3742200, 48648600, -340540200, 0, 40864824000, -694702008000, 6252318072000, 0, -1187940433680000, 24946749107280000, -274414240180080000, 0, 75738330289702080000, -1893458257242552000000, 24614957344153176000000, 0

Offset

4,2

Comments

d^ny/dx^n = (((-1)^(n-1))*(n-1)!)*sin(n*arctan(1/x)) /(1+x^2)^(n/2) - (proof by recurrence). If n = 1, 2, 3, the values of the derivatives at x=1 are respectively 1/2, -1/2, 1/2.

d^ny/dx^n = n!*sum(k=1..n, (binomial(n-1,k-1)*(-1)^(n-k)*x^(n-k)*(1+x^2)^(-n)*(-1)^((k-1)/2)*(1+(-1)^(k-1)))/(2*k)). - Vladimir Kruchinin, Apr 22 2011

Links

Alois P. Heinz, Table of n, a(n) for n = 4..400

Formula

a(n) = (((-1)^(n-1))*(n-1)!)*sin(n*arctan(1))/2^(n/2).

a(n) = 2^(-n-1)*n!*sum(k=1..n, (((-1)^(k-1)+1)*(-1)^(n-k+(k-1)/2)*binomial(n-1,k-1))/k). - Vladimir Kruchinin, Apr 22 2011

abs(a(n)) = abs(integrate(x=0..infty, sin(x)*exp(-x)*x^(n-1))) (see Mathematica code below). - John M. Campbell, Jun 21 2011

E.g.f.: arctan(x+1). - Alois P. Heinz, Feb 14 2015

Example

a(5) = -3 because d^5y/dx^5 = 384*x^4/(1 + x^2)^5 - 288*x^2/(1 + x^2)^4 + 24/(1 + x^2)^3, and for x=1 we obtain 384/32 - 288/16 + 24/8 = -3.

Maple

# First program, with the formula:
n0:= 50: T:=array(1..n0+1):for n from 1 to n0 do:T[n]:=(((-1)^(n-1))*(n-1) !)*sin(n*arctan(1)) /(2^(n/2)):od:print(T):
# Second program, with the Maple instruction D(f):
n0:= 50: T:=array(1..n0+1):f:=x->arctan(x):for n from 1 to n0 do:D(f): T[n]:=(D(f)(1)):f:=D(f):od: print(T):
# third Maple program:
a:= n-> n!*coeff(series(arctan(x+1), x, n+1), x, n):
seq(a(n), n=4..40);  # Alois P. Heinz, Feb 14 2015

Mathematica

Table[Abs[Integrate[Sin[x]*E^(-x)*(x^(n - 1)), {x, 0, Infinity}]], {n, 4, 28}] (* John M. Campbell, Jun 21 2011 *)

Prog

(Maxima)
a(n):=2^(-n-1)*n!*sum((((-1)^(k-1)+1)*(-1)^(n-k+(k-1)/2)*binomial(n-1, k-1))/k, k, 1, n); /* Vladimir Kruchinin, Apr 22 2011 */

Crossrefs

Cf. A005359 (n-th derivatives of arctan(x) at x = 0).

Sequence in context: A101165 A127407 A196237 * A161400 A112810 A334078

Adjacent sequences: A177143 A177144 A177145 * A177147 A177148 A177149

Keyword

sign

Author

Michel Lagneau, May 03 2010

Status

approved