This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A098430 #99 Aug 27 2025 03:58:19
%S A098430 1,8,96,1280,17920,258048,3784704,56229888,843448320,12745441280,
%T A098430 193730707456,2958796259328,45368209309696,697972450918400,
%U A098430 10768717814169600,166556168859156480,2581620617316925440,40091049586568724480,623638549124402380800,9715632133727531827200
%N A098430 a(n) = 4^n*(2*n)!/(n!)^2.
%C A098430 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 A098430 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 A098430 Hankel transform is A121913. - _Philippe Deléham_, Mar 01 2009
%C A098430 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 A098430 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 A098430 Number of orthogonal pairs of vectors of length 2n, constructed with any symmetric binary-valued symbol set. - _Ross Drewe_, May 18 2018
%C A098430 Diagonal of the rational function 1 / (1 - 4*x - y). - _Ilya Gutkovskiy_, Apr 24 2025
%H A098430 Vincenzo Librandi, <a href="/A098430/b098430.txt">Table of n, a(n) for n = 0..200</a>
%H A098430 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 A098430 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 A098430 a(n) = 4^n*binomial(2*n, n) = 4^n*A000984(n).
%F A098430 E.g.f.: exp(8*x)*BesselI(0, 8*x).
%F A098430 G.f.: 1/sqrt(1-16*x). - _Zerinvary Lajos_, Dec 20 2008, corrected _R. J. Mathar_, May 18 2009
%F A098430 a(n) = (1/Pi)*Integral_{x=-2..2} (2*x)^(2*n)/sqrt((2-x)*(2+x)) dx. - _Peter Luschny_, Sep 12 2011
%F A098430 D-finite with recurrence: n*a(n) + 8*(-2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Nov 10 2014
%F A098430 a(n) = A249308(2*n). - _Reinhard Zumkeller_, Nov 14 2014
%F A098430 a(n) = 16^n*hypergeometric([-2*n, 1/2], [1], 2). - _Peter Luschny_, May 19 2015
%F A098430 a(n) = A174301(2n,n). - _Alois P. Heinz_, Apr 15 2019
%F A098430 From _Amiram Eldar_, Jul 21 2020: (Start)
%F A098430 Sum_{n>=0} 1/a(n) = 16/15 + 16*sqrt(15)*arcsin(1/4)/225.
%F A098430 Sum_{n>=0} (-1)^n/a(n) = 16/17 - 16*sqrt(17)*arcsinh(1/4)/289. (End)
%F A098430 a(n) = Sum_{k = 0..2*n} (-1)^k *A000984(k) * A000984(2*n-k). Cf. Sum_{k = 0..2*n} A000984(k) * A000984(2*n-k) = 16^n. - _Peter Bala_, Aug 23 2025
%p A098430 A098430 := n -> 4^n*binomial(2*n,n): seq(A098430(n), n=0..30); # _Wesley Ivan Hurt_, Jul 16 2014
%t A098430 CoefficientList[Series[1/Sqrt[1 - 16 x], {x, 0, 16}], x] (* _Robert G. Wilson v_, Jun 28 2012 *)
%t A098430 Table[4^n(2n)!/(n!)^2,{n,0,20}] (* _Harvey P. Dale_, Aug 13 2021 *)
%o A098430 (PARI) /* as lattice paths: same as in A092566 but use */
%o A098430 steps=[[1,0], [1,0], [0,1], [0,1]]; /* note the double [1,0] and [0,1] */
%o A098430 /* _Joerg Arndt_, Jul 01 2011 */
%o A098430 (Magma) [4^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // _Vincenzo Librandi_, Jul 05 2011
%o A098430 (Haskell)
%o A098430 a098430 n = a000302 n * a000984 n -- _Reinhard Zumkeller_, Nov 14 2014
%o A098430 (Sage)
%o A098430 a = lambda n: 16^n*hypergeometric([-2*n, 1/2], [1], 2)
%o A098430 [simplify(a(n)) for n in range(23)] # _Peter Luschny_, May 19 2015
%Y A098430 Cf. A000302, A000984, A174301, A249308.
%K A098430 easy,nonn,changed
%O A098430 0,2
%A A098430 _Paul Barry_, Sep 07 2004