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],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)
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
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 *)
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
KEYWORD
sign,nice,easy
AUTHOR
N. J. A. Sloane, Dec 11 1996
EXTENSIONS
Additional comments from Michael Somos, Dec 13 2002
STATUS
approved