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A034428
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E.g.f.: 1 - (1-x)*(tan(x) + sec(x)).
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6
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0, 0, 1, 1, 3, 9, 35, 155, 791, 4529, 28839, 201939, 1542739, 12767689, 113794603, 1086657403, 11068604847, 119790363489, 1372696498127, 16603828720547, 211406514019115, 2826296899863929, 39584082775592211, 579600224535319371
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OFFSET
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0,5
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COMMENTS
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Also: number of permutations on n elements having the descent pattern: up, up, down, up, down, ...; a(n) = n * E_{n-1} - E_{n} where E_{n} denotes the Euler numbers, see sequence A000111. - Richard Ehrenborg, Feb 12 2002
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REFERENCES
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R. Ehrenborg and S. Mahajan, Maximizing the descent statistic, Annals Combin., 2 (1998), no. 2, 111-129.
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LINKS
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Peter J. Cameron and Liam Stott, Trees and cycles, arXiv:2010.14902 [math.CO], 2020. See p. 25.
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FORMULA
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E.g.f.: 1 - (1-x)*(tan(x) + sec(x)).
E.g.f.: E(x) = x + x*(x-1)/U(0) where U(k) = 4k + 1 - x/(2 - x/(4k + 3 + x/(2 + x/U(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2012
E.g.f.: x + 2*x*(x-1)/(U(0)-x) where U(k) = 4*k+2 - x^2/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 31 2013
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MATHEMATICA
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With[{nn=30}, Drop[CoefficientList[Series[1-(1-x)(Tan[x]+Sec[x]), {x, 0, nn}], x]Range[0, nn]!, 2]] (* Harvey P. Dale, Jan 22 2012 *)
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PROG
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(PARI) a(n)=n!*polcoeff(1-(1-x)*(tan(x+x*O(x^n))+1/cos(x+x*O(x^n))), n)
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CROSSREFS
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Essentially the same as A131281(n)/2.
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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STATUS
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approved
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