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Search: id:A002420
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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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+0
21
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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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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 (pbarry(AT)wit.ie), Sep 18 2008]
Hankel transform is (2n+1)*(-2)^n or (-1)^n*A014480. [From Paul Barry (pbarry(AT)wit.ie), Jan 22 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), 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)
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
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 (pbarry(AT)wit.ie), Jan 22 2009]
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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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PROGRAM
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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.
A000984 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 02 2009]
Sequence in context: A025244 A132824 A078801 this_sequence A112556 A054100 A034165
Adjacent sequences: A002417 A002418 A002419 this_sequence 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 (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments from Michael Somos, Dec 13 2002
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