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A168491
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(-1)^n*Catalan(n).
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7
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1, -1, 2, -5, 14, -42, 132, -429, 1430, -4862, 16796, -58786, 208012, -742900, 2674440, -9694845, 35357670, -129644790, 477638700, -1767263190, 6564120420, -24466267020, 91482563640, -343059613650, 1289904147324, -4861946401452, 18367353072152, -69533550916004
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OFFSET
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0,3
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COMMENTS
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Second inverse binomial transform of A001405. Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...].
Also the expansion of real root of y+y^2=x, With offset 1, series reversion of x+x^2. [Robert G. Wilson v]
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LINKS
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Table of n, a(n) for n=0..27.
Index to sequences related to reversion of series
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FORMULA
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a(n) = (-1)^n*A000108(n). G.f.: (sqrt(1+4x)-1)/2x.
E.g.f.: exp(-2*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Peter Luschny, Aug 26 2012
(n+1)*a(n) +2*(2*n-1)*a(n-1)=0. - R. J. Mathar, Oct 06 2012
G.f.: 1 / (1 + x / (1 + x / (1 + x / ...))). - Michael Somos, Jan 03 2013
G.f.: 1/(x*Q(0)) - 1/x, where Q(k)= 1 - (4*k+1)*x/(k+1 - x*(2*k+2)*(4*k+3)/(2*x*(4*k+3) - (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
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EXAMPLE
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1 - x + 2*x^2 - 5*x^3 + 14*x^4 - 42*x^5 + 132*x^6 - 429*x^7 +- ...
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MATHEMATICA
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CoefficientList[InverseSeries[Series[y + y^2, {y, 0, 28}], x]/x, x] (* Robert G. Wilson v *)
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PROG
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(PARI) a(n)=(-1)^n*binomial(2*n, n)/(n+1); \\ Joerg Arndt, May 15 2013
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CROSSREFS
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Sequence in context: A211216 A115140 A120588 * A000108 A057413 A126567
Adjacent sequences: A168488 A168489 A168490 * A168492 A168493 A168494
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KEYWORD
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sign,less,changed
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AUTHOR
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Philippe DELEHAM, Nov 27 2009
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STATUS
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approved
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