A002420 Expansion of sqrt(1 - 4*x) in powers of x.

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

Offset: 0

Author

N. J. A. Sloane, Dec 11 1996

Keywords

• sign
• nice
• easy

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

Examples

			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 - ...
		

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.
• R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502.0097 (2025) Table 4
• 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.

Crossrefs

A068875 and A262543 are essentially the same sequence as this.

Cf. A000108, A000984, A001622.

Programs

Magma

[Binomial(2*n, n)/(1-2*n): n in [0..30]]; // G. C. Greubel, Aug 12 2018

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-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 *)

PARI

{a(n) = binomial(2*n, n) / (1 - 2*n)} /* Michael Somos, Jul 12 2008 */

Sage

[catalan_number(n)*((1+n)/(1-2*n)) for n in range(30)] # G. C. Greubel, Nov 26 2018

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],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)

From Peter Bala, Mar 31 2024: (Start)

a(n) = (4^n)*Sum_{k = 0..2*n} (-1)^k*binomial(1/2, k)*binomial(1/2, 2*n - k).

(4^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).

(1/2)*Sum_{k = 0..n} a(k)*a(2*n-k) = (Catalan(n-1))^2 = A001246(n) for n >= 1.

Sum_{k = 0..2*n} a(k)*a(2*n-k) = 0 for n >= 1. (End)

Extensions

Additional comments from Michael Somos, Dec 13 2002