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A002420 Expansion of sqrt(1 - 4*x) in powers of x.
(Formerly M0337 N0128)
33
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.

P.-Y. Huang, S.-C. Liu, Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.

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

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.

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. [Annotated scanned copy]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 411

N. J. A. Sloane, Notes on A984 and A2420-A2424

FORMULA

G.f.: sqrt(1-4x) = 1F0(-1/2;;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.: 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

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

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],1). - Peter Luschny, Apr 26 2016

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[0] = 1; Table[a[n], {n, 0, 27}] (* Jean-Fran├žois 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 */

CROSSREFS

A068875 and A262543 are essentially the same sequence as this.

Cf. A000108, A000984.

Sequence in context: A025244 A132824 A078801 * 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 July 20 16:53 EDT 2017. Contains 289628 sequences.