A098430 - OEIS

%I #88 Aug 30 2024 02:55:47

%S 1,8,96,1280,17920,258048,3784704,56229888,843448320,12745441280,

%T 193730707456,2958796259328,45368209309696,697972450918400,

%U 10768717814169600,166556168859156480,2581620617316925440,40091049586568724480,623638549124402380800,9715632133727531827200

%N a(n) = 4^n*(2*n)!/(n!)^2.

%C a(n) counts walks of 2n steps North, East, South or West that start at the origin and end on the line y=x. For example, a(1)=8 counts EW, EN, NE, NS, WE, WS, SN, SW. If the walk has i East and j North steps, then it must have n-j West and n-i South steps. There are Multinomial[i,j,n-j,n-i] ways to arrange these steps and summing over i and j gives the result. - _David Callan_, Oct 11 2005

%C Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), both of two kinds. - _Joerg Arndt_, Jul 01 2011

%C Hankel transform is A121913. - _Philippe Deléham_, Mar 01 2009

%C Convolving a(n) with itself yields A001025, the powers of 16. Thus the limiting ratio of this sequence is 16. - _Bob Selcoe_, Jul 16 2014

%C Number of strings x of length 4n over the alphabet {1, -1} such that the dot product of x with (x reversed) is 0. - _Jeffrey Shallit_, Mar 06 2017

%C Number of orthogonal pairs of vectors of length 2n, constructed with any symmetric binary-valued symbol set. - _Ross Drewe_, May 18 2018

%H Vincenzo Librandi, <a href="/A098430/b098430.txt">Table of n, a(n) for n = 0..200</a>

%H Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

%F a(n) = 4^n*binomial(2*n, n) = 4^n*A000984(n).

%F E.g.f.: exp(8*x)*BesselI(0, 8*x).

%F G.f.: 1/sqrt(1-16*x). - _Zerinvary Lajos_, Dec 20 2008, corrected _R. J. Mathar_, May 18 2009

%F a(n) = (1/Pi)*Integral_{x=-2..2} (2*x)^(2*n)/sqrt((2-x)*(2+x)) dx. - _Peter Luschny_, Sep 12 2011

%F D-finite with recurrence: n*a(n) + 8*(-2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Nov 10 2014

%F a(n) = A249308(2*n). - _Reinhard Zumkeller_, Nov 14 2014

%F a(n) = 16^n*hypergeometric([-2*n, 1/2], [1], 2). - _Peter Luschny_, May 19 2015

%F a(n) = A174301(2n,n). - _Alois P. Heinz_, Apr 15 2019

%F From _Amiram Eldar_, Jul 21 2020: (Start)

%F Sum_{n>=0} 1/a(n) = 16/15 + 16*sqrt(15)*arcsin(1/4)/225.

%F Sum_{n>=0} (-1)^n/a(n) = 16/17 - 16*sqrt(17)*arcsinh(1/4)/289. (End)

%p A098430 := n -> 4^n*binomial(2*n,n): seq(A098430(n), n=0..30); # _Wesley Ivan Hurt_, Jul 16 2014

%t CoefficientList[Series[1/Sqrt[1 - 16 x], {x, 0, 16}], x] (* _Robert G. Wilson v_, Jun 28 2012 *)

%t Table[4^n(2n)!/(n!)^2,{n,0,20}] (* _Harvey P. Dale_, Aug 13 2021 *)

%o (PARI) /* as lattice paths: same as in A092566 but use */

%o steps=[[1,0], [1,0], [0,1], [0,1]]; /* note the double [1,0] and [0,1] */

%o /* _Joerg Arndt_, Jul 01 2011 */

%o (Magma) [4^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // _Vincenzo Librandi_, Jul 05 2011

%o (Haskell)

%o a098430 n = a000302 n * a000984 n -- _Reinhard Zumkeller_, Nov 14 2014

%o (Sage)

%o a = lambda n: 16^n*hypergeometric([-2*n, 1/2], [1], 2)

%o [simplify(a(n)) for n in range(23)] # _Peter Luschny_, May 19 2015

%Y Cf. A000302, A000984, A174301, A249308.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Sep 07 2004