|
%I #35 Sep 16 2017 03:37:27
%S 0,1,-1,0,6,-30,90,0,-2520,22680,-113400,0,7484400,-97297200,
%T 681080400,0,-81729648000,1389404016000,-12504636144000,0,
%U 2375880867360000,-49893498214560000,548828480360160000,0,-151476660579404160000
%N Exponential generating function is tanh(log(1+x)).
%H Brandon Humpert and Jeremy L. Martin, <a href="https://dmtcs.episciences.org/2930/">The Incidence Hopf Algebra of Graphs</a>, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), pp. 517-526. [See Example 3.4]
%F a(0) = 0, a(4n+3) = 0, a(n) = (-1)^[n == 2, 5, 8 mod 8] * n!/2^floor(n/2). - _Ralf Stephan_, Mar 06 2004
%F From _Peter Bala_, Nov 25 2011: (Start)
%F (1): a(n) = i*n!/2^(n+1)*{(i-1)^(n+1)-(-1-i)^(n+1)} for n>=1.
%F The function tanh(log(1+x)) is a disguised form of the rational function (x^2+2*x)/(x^2+2*x+2). Observe that
%F (2): (x^2+2*x)/(x^2+2*x+2) = d/dx[x - atan((x^2+2*x)/(2*x+2))].
%F Hence, with an offset of 1, the egf for this sequence is
%F (3): x - atan((x^2+2*x)/(2*x+2)) = x^2/2! - x^3/3! + 6*x^5/5!- 30*x^6/6! + 90*x^7/7! - ....
%F This sequence is closely related to the series reversion of the function E(x)-1, where E(x) = sec(x)+tan(x) is the egf for the sequence of zigzag numbers A000111. Under the change of variable x -> sec(x)+tan(x)-1 the rational function (x^2+2*x)/(2*x+2) transforms to tan(x). Hence atan((x^2+2*x)/(2*x+2)) is the inverse function of sec(x)+tan(x)-1.
%F Recurrence relation:
%F (4): 2*a(n)+2*n*a(n-1)+n*(n-1)*a(n-2) = 0 with a(1) = 1, a(2) = -1.
%F (End)
%o (PARI) A009775(n)=polcoeff(tanh(log(1+x+O(x^n)*x)),n)*n! \\ _M. F. Hasler_, Oct 10 2012
%Y Cf. A052277, A007019, A046979, A007415, A007452, A092820, A217260.
%K sign,easy
%O 0,5
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
|
