cp's OEIS Frontend

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

%I A002420 M0337 N0128 #122 Feb 28 2025 06:22:39
%S A002420 1,-2,-2,-4,-10,-28,-84,-264,-858,-2860,-9724,-33592,-117572,-416024,
%T A002420 -1485800,-5348880,-19389690,-70715340,-259289580,-955277400,
%U A002420 -3534526380,-13128240840,-48932534040,-182965127280,-686119227300,-2579808294648,-9723892802904,-36734706144304
%N A002420 Expansion of sqrt(1 - 4*x) in powers of x.
%C A002420 Also expansion of complementary modulus k' in powers of m/4 = k^2/4.
%C A002420 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.
%C A002420 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
%C A002420 Hankel transform is (2n+1)*(-2)^n or (-1)^n*A014480. - _Paul Barry_, Jan 22 2009
%C A002420 Equals polcoeff inverse of A000984. - _Gary W. Adamson_, Jun 02 2009
%C A002420 |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
%D A002420 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
%D A002420 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.
%D A002420 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002420 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002420 T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.
%H A002420 T. D. Noe, <a href="/A002420/b002420.txt">Table of n, a(n) for n=0..200</a>
%H A002420 Alexander Barg, <a href="https://arxiv.org/abs/2005.12995">Stolarsky's invariance principle for finite metric spaces</a>, arXiv:2005.12995 [math.CO], 2020.
%H A002420 S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="http://dx.doi.org/10.1021/ci00026a012">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., Vol. 35 (1995), pp. 743-751.
%H A002420 S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="/A002057/a002057.pdf">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., Vol. 35 (1995), pp. 743-751. [Annotated scanned copy]
%H A002420 P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p45">Congruences of Finite Summations of the Coefficients in certain Generating Functions</a>, The Electronic Journal of Combinatorics, Vol. 21, No. 2 (2014), Article P2.45.
%H A002420 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=411">Encyclopedia of Combinatorial Structures 411</a>.
%H A002420 R. J. Mathar, <a href="https://vixra.org/abs/2502.0097">The Eggenberger-Polya urn process: Probabilities of revisited ball ratios</a>, vixra:2502.0097 (2025) Table 4
%H A002420 N. J. A. Sloane, <a href="/A000984/a000984.pdf">Notes on A984 and A2420-A2424</a>.
%H A002420 Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%F A002420 G.f.: sqrt(1-4*x) = 1F0(-1/2;;4*x).
%F A002420 a(n) = binomial(2*n, n)/(1-2*n).
%F A002420 a(n) ~ -(1/2)*Pi^(-1/2)*n^(-3/2)*2^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
%F A002420 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
%F A002420 G.f.: 2F1(1,-1/2;1;4x). - _Paul Barry_, Jan 22 2009
%F A002420 a(n) = (-1)^n * binomial(1/2,n)*4^n. - _Vladimir Kruchinin_, May 22 2011
%F A002420 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
%F A002420 D-finite with recurrence: n*a(n) +2*(3-2*n)*a(n-1)=0. - _R. J. Mathar_, Dec 19 2011
%F A002420 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
%F A002420 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
%F A002420 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
%F A002420 a(n) = 4^n * binomial(n-3/2, -3/2). - _Peter Luschny_, May 06 2014
%F A002420 a(n) = 4^n*hypergeom([-n,3/2],[1],1). - _Peter Luschny_, Apr 26 2016
%F A002420 From _Amiram Eldar_, Mar 24 2022: (Start)
%F A002420 Sum_{n>=0} 1/a(n) = -2*Pi/(9*sqrt(3)).
%F A002420 Sum_{n>=0} (-1)^n/a(n) = 32/25 - 12*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). (End)
%F A002420 From _Peter Bala_, Mar 31 2024: (Start)
%F A002420 a(n) = (4^n)*Sum_{k = 0..2*n} (-1)^k*binomial(1/2, k)*binomial(1/2, 2*n - k).
%F A002420 (4^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
%F A002420 (1/2)*Sum_{k = 0..n} a(k)*a(2*n-k) = (Catalan(n-1))^2 = A001246(n) for n >= 1.
%F A002420 Sum_{k = 0..2*n} a(k)*a(2*n-k) = 0 for n >= 1. (End)
%e A002420 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 - ...
%p A002420 A002420:=n->binomial(2*n, n)/(1-2*n); seq(A002420(n), n=1..30); # _Wesley Ivan Hurt_, May 08 2014
%t A002420 a[n_] := -2n(2n-2)! / n!^2; a[0] = 1; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Dec 07 2011 *)
%t A002420 Table[If[n==0,1,-2 CatalanNumber[n-1]], {n,0,27}] (* _Peter Luschny_, Feb 27 2017 *)
%t A002420 CoefficientList[Series[Sqrt[1-4x],{x,0,30}],x] (* _Harvey P. Dale_, Jul 04 2017 *)
%o A002420 (PARI) {a(n) = binomial(2*n, n) / (1 - 2*n)} /* _Michael Somos_, Jul 12 2008 */
%o A002420 (Magma) [Binomial(2*n, n)/(1-2*n): n in [0..30]]; // _G. C. Greubel_, Aug 12 2018
%o A002420 (Sage) [catalan_number(n)*((1+n)/(1-2*n)) for n in range(30)] # _G. C. Greubel_, Nov 26 2018
%Y A002420 A068875 and A262543 are essentially the same sequence as this.
%Y A002420 Cf. A000108, A000984, A001622.
%K A002420 sign,nice,easy
%O A002420 0,2
%A A002420 _N. J. A. Sloane_, Dec 11 1996
%E A002420 Additional comments from _Michael Somos_, Dec 13 2002