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A104624
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Expansion of exp( asinh( -2*x ) ) in powers of x.
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2
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1, -2, 2, 0, -2, 0, 4, 0, -10, 0, 28, 0, -84, 0, 264, 0, -858, 0, 2860, 0, -9724, 0, 33592, 0, -117572, 0, 416024, 0, -1485800, 0, 5348880, 0, -19389690, 0, 70715340, 0, -259289580, 0, 955277400, 0, -3534526380, 0, 13128240840, 0, -48932534040, 0, 182965127280, 0, -686119227300, 0
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OFFSET
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0,2
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COMMENTS
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First column in inverse of A054335.
With offset 1 the coefficient sequence of series reversion of A000984 (binomial(2n,n)) also with offset 1. [Michael Somos, Jan 14 2011]
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LINKS
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Table of n, a(n) for n=0..49.
Vladimir Kruchinin, Compositae and their properties , arXiv:1103.2582
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FORMULA
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G.f.: sqrt( 1 + 4*x^2 ) - 2*x = exp( asinh( -2*x ) ). [Michael Somos, Jan 14 2011]
The positive sequence 1,2,2,0,2,.... has g.f. 2(1+x)-sqrt(1-4x^2). - Paul Barry, Oct 10 2007
From Vladimir Kruchinin, Jan 16 2011 (Start)
The o.g.f. A(x) satisfies A(x)=x*sqrt(1-4*A(x)),
a(n) = 1/(n*(n+1)) * sum(j=0...n+1, j * 2^(j) * binomial(2*n-j-1,n-1) * binomial(n+1,j) * (-1)^(n-j)). (End)
Conjecture: n*a(n) +(n-1)*a(n-1) +4*(n-3)*a(n-2) +4*(n-4)*a(n-3)=0. - R. J. Mathar, Nov 15 2012
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EXAMPLE
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1 - 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + 264*x^14 + ...
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MATHEMATICA
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CoefficientList[ Series[ Exp[ ArcSinh[ -2x]], {x, 0, 49}], x]
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PROG
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{a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) - 2*x, n ) )} /* Michael Somos, Jan 14 2011 */
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CROSSREFS
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Sequence in context: A080378 A120439 A182122 * A193863 A213209 A049850
Adjacent sequences: A104621 A104622 A104623 * A104625 A104626 A104627
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KEYWORD
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easy,sign
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AUTHOR
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Paul Barry, Mar 17 2005
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STATUS
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approved
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