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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS 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 EXTENSIONS Extended with signs by Olivier Gérard, Mar 15 1997 STATUS approved

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Last modified April 12 18:58 EDT 2021. Contains 342932 sequences. (Running on oeis4.)