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A104624
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Expansion of exp( asinh( -2*x ) ) in powers of x.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
| Vladimir Kruchinin, Compositae and their properties , arXiv:1103.2582
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FORMULA
| 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 (pbarry(AT)wit.ie), Oct 10 2007
From Vladimir Kruchinin (kru(AT)ie.tusur.ru), 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)
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EXAMPLE
| 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
| CoefficientList[ Series[ Exp[ ArcSinh[ -2x]], {x, 0, 49}], x]
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PROG
| {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
| Sequence in context: A122157 A080378 A120439 * A193863 A049850 A050949
Adjacent sequences: A104621 A104622 A104623 * A104625 A104626 A104627
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KEYWORD
| easy,sign
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 17 2005
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