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A002420
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Expansion of sqrt(1 - 4*x) in powers of x.
(Formerly M0337 N0128)
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25
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1, -2, -2, -4, -10, -28, -84, -264, -858, -2860, -9724, -33592, -117572, -416024, -1485800, -5348880, -19389690, -70715340, -259289580, -955277400, -3534526380, -13128240840, -48932534040, -182965127280, -686119227300, -2579808294648, -9723892802904, -36734706144304
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OFFSET
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0,2
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COMMENTS
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Also expansion of complementary modulus k' in powers of m/4=k^2/4.
Series reversion of x(Sum_{k>=0} a(k)x^(2k)) is x(Sum_{k>=0} C(2k)x^(2k)) where C() is Catalan numbers A000108.
The g.f of the reciprocal sequence 1,-1/2,-1/2,... is F(1,1;-1/2;x/4). [From Paul Barry, Sep 18 2008]
Hankel transform is (2n+1)*(-2)^n or (-1)^n*A014480. [From Paul Barry, Jan 22 2009]
Contribution from Gary W. Adamson, Jun 02 2009: (Start)
Equals polcoeff inverse of A000984. Note: the convolution square of A000984
equals the powers of 4: (1, 4, 16, 64,...). (End)
|a(n)| is the number of lattice paths in steps of (1,1) and (1,-1) that begin at the origin and end at (2n,0) but otherwise never touch (or cross) the x axis. Note the paths are in both the first and fourth quadrants. O.g.f. is 2xC(x)+1 where C(x) is the o.g.f. for A000108 (Catalan numbers). - Geoffrey Critzer, Jan 17 2012
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REFERENCES
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J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 411
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FORMULA
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G.f.: sqrt(1-4x). a(n)=binomial(2*n, n)/(1-2*n).
a(n) ~ -(1/2)*pi^(-1/2)*n^(-3/2)*2^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
0 = 16 * a(n) * a(k) * a(n+k+1) - 8 * a(n) * a(k) * a(n+k+2) + a(n+1) * a(k) * a(n+k+2) - a(n+1) * a(k+1) * a(n+k+1) + a(n) * a(k+1) * a(n+k+2) for all n and k. - Michael Somos, Jul 12 2008
G.f.: F(1,-1/2;1;4x). [From Paul Barry, Jan 22 2009]
a(n)=(-1)^n*binomial(1/2,n)*4^n. [From Vladimir Kruchinin, May 22 2011]
G.f.: A(x)=(1-4*x)^(1/2)=1 - 2*x - 2*x^2/G(0) ; G(k) = 1 - 2*x - x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011
n*a(n) +2*(3-2n)*a(n-1)=0. - R. J. Mathar, Dec 19 2011
E.g.f.: a(n) = (-1)^n*n!* [x^n] exp(-2*x)*((1 + 4*x)*BesselI(0, 2*x) + 4*x*BesselI(1, 2*x)). -Peter Luschny, Aug 25 2012
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EXAMPLE
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sqrt(1 - 4*x) = 1 - 2*x - 2*x^2 - 4*x^3 - 10*x^4 - 28*x^5 - 84*x^6 - 264*x^7 - 858*x^8 - 2860*x^9 - ...
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MATHEMATICA
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a[n_] := -2n(2n-2)! / n!^2; a[0] = 1; Table[a[n], {n, 0, 27}] (* From Jean-François Alcover, Dec 07 2011 *)
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PROG
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(PARI) {a(n) = binomial(2*n, n) / (1 - 2*n)} /* Michael Somos Jul 12 2008 */
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CROSSREFS
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Cf. A000108.
Cf. A000984. [From Gary W. Adamson, Jun 02 2009]
Sequence in context: A025244 A132824 A078801 * A112556 A054100 A034165
Adjacent sequences: A002417 A002418 A002419 * A002421 A002422 A002423
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KEYWORD
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sign,nice,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Additional comments from Michael Somos, Dec 13 2002
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STATUS
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approved
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