

A009775


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


11



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
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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. 517526. [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)*{(i1)^(n+1)(1i)^(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(n1)+n*(n1)*a(n2) = 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



