A217260 Expansion of e.g.f. 2*arctan(1+x) - Pi/2.

1, -1, 1, 0, -6, 30, -90, 0, 2520, -22680, 113400, 0, -7484400, 97297200, -681080400, 0, 81729648000, -1389404016000, 12504636144000, 0, -2375880867360000, 49893498214560000, -548828480360160000

Offset: 1

Author

Vladimir Kruchinin, Mar 17 2013

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 1..477

Crossrefs

Cf. A000111 (e.g.f. sec(x)+tan(x)), A009775.

Programs

Maple

seq(2^(1-n/2)*sin(3/4*Pi*n)*(n-1)!, n=1..50); # Robert Israel, Jan 17 2017

Mathematica

Table[2^(1 - n/2)*(n - 1)!*Sin[3*Pi*n/4], {n, 30}] (* Wesley Ivan Hurt, Oct 14 2023 *)

Maxima

a(n):=n!*sum(((-1)^(n+i)*binomial(n-1,2*i-2))/(2*i-1),i,1,(n+1)/2)/2^(n-1);

Formula

E.g.f.: 2*arctan(1+x) - Pi/2.

a(n) = n!*(Sum_{i=1..floor(n+1)/2} ((-1)^(n+i)*binomial(n-1, 2*i-2))/(2*i-1))/2^(n-1).

E.g.f. is the series reversion of sec(x) + tan(x) - 1.

From Robert Israel, Jan 17 2017: (Start)

a(n) = (n-1)*a(n-1) - (n-1)*(n-2)*a(n-2)/2.

a(n) = 2^(1-n/2)*(n-1)!*sin(3*Pi*n/4). (End)