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 A002420 Expansion of sqrt(1 - 4*x) in powers of x. (Formerly M0337 N0128) 46
 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; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). - Paul Barry, Sep 18 2008 Hankel transform is (2n+1)*(-2)^n or (-1)^n*A014480. - Paul Barry, Jan 22 2009 Equals polcoeff inverse of A000984. - Gary W. Adamson, Jun 02 2009 |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 REFERENCES J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8. 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. LINKS T. D. Noe, Table of n, a(n) for n=0..200 Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020. 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., Vol. 35 (1995), pp. 743-751. 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., Vol. 35 (1995), pp. 743-751. [Annotated scanned copy] P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, Vol. 21, No. 2 (2014), Article P2.45. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 411. N. J. A. Sloane, Notes on A984 and A2420-A2424. Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021. FORMULA G.f.: sqrt(1-4*x) = 1F0(-1/2;;4*x). 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.: 2F1(1,-1/2;1;4x). - Paul Barry, Jan 22 2009 a(n) = (-1)^n * binomial(1/2,n)*4^n. - 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 D-finite with recurrence: n*a(n) +2*(3-2*n)*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 G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013 G.f.: 2*G(0) - 1, where G(k) = 2*x*(2*k+1) + (k+1) - 2*x*(k+1)*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 02 2013 a(n) = 4^n * binomial(n-3/2, -3/2). - Peter Luschny, May 06 2014 a(n) = 4^n*hypergeom([-n,3/2],,1). - Peter Luschny, Apr 26 2016 From Amiram Eldar, Mar 24 2022: (Start) Sum_{n>=0} 1/a(n) = -2*Pi/(9*sqrt(3)). Sum_{n>=0} (-1)^n/a(n) = 32/25 - 12*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). (End) EXAMPLE 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 - ... MAPLE A002420:=n->binomial(2*n, n)/(1-2*n); seq(A002420(n), n=1..30); # Wesley Ivan Hurt, May 08 2014 MATHEMATICA a[n_] := -2n(2n-2)! / n!^2; a = 1; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Dec 07 2011 *) Table[If[n==0, 1, -2 CatalanNumber[n-1]], {n, 0, 27}] (* Peter Luschny, Feb 27 2017 *) CoefficientList[Series[Sqrt[1-4x], {x, 0, 30}], x] (* Harvey P. Dale, Jul 04 2017 *) PROG (PARI) {a(n) = binomial(2*n, n) / (1 - 2*n)} /* Michael Somos, Jul 12 2008 */ (Magma) [Binomial(2*n, n)/(1-2*n): n in [0..30]]; // G. C. Greubel, Aug 12 2018 (Sage) [catalan_number(n)*((1+n)/(1-2*n)) for n in range(30)] # G. C. Greubel, Nov 26 2018 CROSSREFS A068875 and A262543 are essentially the same sequence as this. Cf. A000108, A000984, A001622. Sequence in context: A339830 A078801 A309159 * A284016 A112556 A254400 Adjacent sequences:  A002417 A002418 A002419 * A002421 A002422 A002423 KEYWORD sign,nice,easy AUTHOR N. J. A. Sloane, Dec 11 1996 EXTENSIONS Additional comments from Michael Somos, Dec 13 2002 STATUS approved

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Last modified October 6 10:31 EDT 2022. Contains 357263 sequences. (Running on oeis4.)