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A012816
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E.g.f. arctan(sec(x)*sinh(x)) (odd powers only).
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7
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1, 2, -20, -488, 22160, 1616672, -172976960, -25518205568, 4964227109120, 1231298393825792, -379260096755225600, -142026494757146421248, 63547531933929827962880, 33481297996129270926221312, -20517021964757071715832381440, -14468510293983989090015078678528
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OFFSET
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0,2
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COMMENTS
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The unsigned sequence {|a(n)|}n>=1 = [2,20,488,22160,...] enumerates binary increasing trees on 2*n vertices with a perfect matching (Kuba and Wagner).
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LINKS
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FORMULA
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1/cosh(x*sqrt(2)) = 1 - 2x^2/2! + 20*x^4/4! - 488*x^6/6! +-...
G.f. (for the unsigned sequence): 1/G(0) where G(k) = 1 - 2*x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013
G.f. (for the unsigned sequence): Q(0), where Q(k) = 1 - 2*x*(k+1)^2/(2*x*(k+1)^2 - 1/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
E.g.f.(for the unsigned sequence, odd powers only): 1 + T(0)*x^2 /(1-x^2), where T(k) = 1 - 2*x^2*(2*k+1)*(2*k+2)/( 2*x^2*(2*k+1)*(2*k+2) + ((2*k+1)*(2*k+2)-2*x^2)*((2*k+3)*(2*k+4)-2*x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013
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EXAMPLE
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arctan(sec(x)*sinh(x)) = x+2/3!*x^3-20/5!*x^5-488/7!*x^7+22160/9!*x^9...
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MAPLE
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a:= n-> (2*n+1)! *
coeff(series(arctan(sec(x)*sinh(x)), x, 2*(n+1)), x, 2*n+1):
seq(a(n), n=0..20);
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MATHEMATICA
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With[{nn=40}, Take[CoefficientList[Series[ArcTan[Sec[x]Sinh[x]], {x, 0, nn}], x] Range[0, nn-1]!, {2, -1, 2}]] (* Harvey P. Dale, Jul 25 2024 *)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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Patrick Demichel (patrick.demichel(AT)hp.com)
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STATUS
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approved
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