%I A002420 M0337 N0128
%S A002420 1,2,2,4,10,28,84,264,858,2860,9724,33592,117572,416024,1485800,5348880,
%T A002420 19389690,70715340,259289580,955277400,3534526380,13128240840,48932534040,
%U A002420 182965127280,686119227300,2579808294648,9723892802904,36734706144304
%V A002420 1,-2,-2,-4,-10,-28,-84,-264,-858,-2860,-9724,-33592,-117572,-416024,-1485800,
               -5348880,
%W A002420 -19389690,-70715340,-259289580,-955277400,-3534526380,-13128240840,-48932534040,
%X A002420 -182965127280,-686119227300,-2579808294648,-9723892802904,-36734706144304
%N A002420 Expansion of sqrt(1 - 4*x) in powers of x.
%C A002420 Also expansion of complementary modulus k' in powers of m/4=k^2/4.
%C A002420 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.
%C A002420 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]
%C A002420 Hankel transform is (2n+1)*(-2)^n or (-1)^n*A014480. [From Paul Barry 
               (pbarry(AT)wit.ie), Jan 22 2009]
%C A002420 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 02 2009: 
               (Start)
%C A002420 Equals polcoeff inverse of A000984. Note: the convolution square of A000984
%C A002420 equals the powers of 4: (1, 4, 16, 64,...). (End)
%D A002420 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002420 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002420 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
%D A002420 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.
%D A002420 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.
%D A002420 T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.
%H A002420 T. D. Noe, <a href="b002420.txt">Table of n, a(n) for n=0..200</a>
%H A002420 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=411">
               Encyclopedia of Combinatorial Structures 411</a>
%F A002420 G.f.: sqrt(1-4x). a(n)=binomial(2*n, n)/(1-2*n).
%F A002420 a(n) ~ -(1/2)*pi^(-1/2)*n^(-3/2)*2^(2*n) - Joe Keane (jgk(AT)jgk.org), 
               Jun 06 2002
%F A002420 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
%F A002420 G.f.: F(1,-1/2;1;4x). [From Paul Barry (pbarry(AT)wit.ie), Jan 22 2009]
%e A002420 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 - ...
%o A002420 (PARI) {a(n) = binomial(2*n, n) / (1 - 2*n)} /* Michael Somos Jul 12 
               2008 */
%Y A002420 Cf. A000108.
%Y A002420 A000984 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 02 2009]
%Y A002420 Sequence in context: A025244 A132824 A078801 this_sequence A112556 A054100 
               A034165
%Y A002420 Adjacent sequences: A002417 A002418 A002419 this_sequence A002421 A002422 
               A002423
%K A002420 sign,nice,easy
%O A002420 0,2
%A A002420 N. J. A. Sloane (njas(AT)research.att.com).
%E A002420 Additional comments from Michael Somos, Dec 13 2002