A009775 Exponential generating function is tanh(log(1+x)).

0, 1, -1, 0, 6, -30, 90, 0, -2520, 22680, -113400, 0, 7484400, -97297200, 681080400, 0, -81729648000, 1389404016000, -12504636144000, 0, 2375880867360000, -49893498214560000, 548828480360160000, 0, -151476660579404160000

Offset

0,5

Links

Table of n, a(n) for n=0..24.

Brandon Humpert and Jeremy L. Martin, The Incidence Hopf Algebra of Graphs, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), pp. 517-526. [See Example 3.4]

Formula

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

From Peter Bala, Nov 25 2011: (Start)

(1): a(n) = i*n!/2^(n+1)*{(i-1)^(n+1)-(-1-i)^(n+1)} for n>=1.

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

(2): (x^2+2*x)/(x^2+2*x+2) = d/dx[x - atan((x^2+2*x)/(2*x+2))].

Hence, with an offset of 1, the egf for this sequence is

(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! - ....

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.

Recurrence relation:

(4): 2*a(n)+2*n*a(n-1)+n*(n-1)*a(n-2) = 0 with a(1) = 1, a(2) = -1.

(End)

Prog

(PARI) A009775(n)=polcoeff(tanh(log(1+x+O(x^n)*x)), n)*n!  \\ M. F. Hasler, Oct 10 2012

Crossrefs

Cf. A052277, A007019, A046979, A007415, A007452, A092820, A217260.

Sequence in context: A055112 A094143 A217260 * A297570 A119536 A107394

Adjacent sequences: A009772 A009773 A009774 * A009776 A009777 A009778

Keyword

sign,easy

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Status

approved