|
| |
|
|
A002420
|
|
Expansion of sqrt(1 - 4*x) in powers of x.
(Formerly M0337 N0128)
|
|
50
|
|
|
|
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
|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
|
|
|
|
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.: 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*hypergeom([-n,3/2],[1],1). - Peter Luschny, Apr 26 2016
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)
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
|
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
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
