cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A128014 Central binomial coefficients C(2n,n) repeated.

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 20, 20, 70, 70, 252, 252, 924, 924, 3432, 3432, 12870, 12870, 48620, 48620, 184756, 184756, 705432, 705432, 2704156, 2704156, 10400600, 10400600, 40116600, 40116600, 155117520, 155117520, 601080390, 601080390
Offset: 0

Views

Author

Paul Barry, Feb 11 2007

Keywords

Comments

Binomial transform is A097893. Hankel transform is A128017.
Hankel transform of a(n+1) is A128018. - Paul Barry, Nov 23 2009
Number of 2n-bead balanced binary necklaces that are equivalent to their reverse. - Andrew Howroyd, Sep 29 2017
Number of ballot sequences of length n in which the vote is tied or decided by 1 vote. - Nachum Dershowitz, Aug 12 2020
Number of binary strings of length n that are abelian squares. - Michael S. Branicky, Dec 21 2020

Crossrefs

Cf. A097893, A128017, A128018.
Cf. A000984, A249095, A249308.

Programs

  • Haskell
    a128014 = a000984 . flip div 2
-- Reinhard Zumkeller, Nov 14 2014
  • Mathematica
    (1+x)/Sqrt[1-4x^2] + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 07 2017 *)
With[{cb=Table[Binomial[2n,n],{n,0,20}]},Riffle[cb,cb]] (* Harvey P. Dale, Feb 17 2020 *)

Formula

G.f.: (1+x)/sqrt(1-4*x^2).
a(n) = C(n,n/2)*(1+(-1)^n)/2 + C(n-1,(n-1)/2)*(1-(-1)^n)/2.
a(n) = (1/Pi)*Integral_{x=-2..2} x^n*(1+x)/(x*sqrt(4-x^2)), as moment sequence.
E.g.f. of a(n+1): Bessel_I(0,2*x)+2*Bessel_I(1,2*x). - Paul Barry, Mar 26 2010
n*a(n) +(n-2)*a(n-1) +4*(-n+1)*a(n-2) +4*(-n+3)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
a(n) = 2^n*Product_{k=0..n-1} ((k/n+1/n)/2)^((-1)^k). - Peter Luschny, Dec 03 2013
From Reinhard Zumkeller, Nov 14 2014: (Start)
a(n) = A000984(floor(n/2)).
a(n) = A249095(n,n) = A249308(n) / 2^n. (End)

A098430 a(n) = 4^n*(2*n)!/(n!)^2.

Original entry on oeis.org

1, 8, 96, 1280, 17920, 258048, 3784704, 56229888, 843448320, 12745441280, 193730707456, 2958796259328, 45368209309696, 697972450918400, 10768717814169600, 166556168859156480, 2581620617316925440, 40091049586568724480, 623638549124402380800, 9715632133727531827200
Offset: 0

Views

Author

Paul Barry, Sep 07 2004

Keywords

Comments

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
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
Hankel transform is A121913. - Philippe Deléham, Mar 01 2009
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
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
Number of orthogonal pairs of vectors of length 2n, constructed with any symmetric binary-valued symbol set. - Ross Drewe, May 18 2018
Diagonal of the rational function 1 / (1 - 4*x - y). - Ilya Gutkovskiy, Apr 24 2025

Links

Crossrefs

Cf. A000302, A000984, A174301, A249308.

Programs

  • Haskell
    a098430 n = a000302 n * a000984 n -- Reinhard Zumkeller, Nov 14 2014
  • Magma
    [4^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011
  • Maple
    A098430 := n -> 4^n*binomial(2*n,n): seq(A098430(n), n=0..30); # Wesley Ivan Hurt, Jul 16 2014
  • Mathematica
    CoefficientList[Series[1/Sqrt[1 - 16 x], {x, 0, 16}], x] (* Robert G. Wilson v, Jun 28 2012 *)
Table[4^n(2n)!/(n!)^2,{n,0,20}] (* Harvey P. Dale, Aug 13 2021 *)
  • PARI
    /* as lattice paths: same as in A092566 but use */
steps=[[1,0], [1,0], [0,1], [0,1]]; /* note the double [1,0] and [0,1] */
/* Joerg Arndt, Jul 01 2011 */
  • Sage
    a = lambda n: 16^n*hypergeometric([-2*n, 1/2], [1], 2)
[simplify(a(n)) for n in range(23)] # Peter Luschny, May 19 2015

Formula

a(n) = 4^n*binomial(2*n, n) = 4^n*A000984(n).
E.g.f.: exp(8*x)*BesselI(0, 8*x).
G.f.: 1/sqrt(1-16*x). - Zerinvary Lajos, Dec 20 2008, corrected R. J. Mathar, May 18 2009
a(n) = (1/Pi)*Integral_{x=-2..2} (2*x)^(2*n)/sqrt((2-x)*(2+x)) dx. - Peter Luschny, Sep 12 2011
D-finite with recurrence: n*a(n) + 8*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 10 2014
a(n) = A249308(2*n). - Reinhard Zumkeller, Nov 14 2014
a(n) = 16^n*hypergeometric([-2*n, 1/2], [1], 2). - Peter Luschny, May 19 2015
a(n) = A174301(2n,n). - Alois P. Heinz, Apr 15 2019
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 16/15 + 16*sqrt(15)*arcsin(1/4)/225.
Sum_{n>=0} (-1)^n/a(n) = 16/17 - 16*sqrt(17)*arcsinh(1/4)/289. (End)
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

A249307 Triangle read by rows: A249095(n,k) * 2^k, k = 0 .. 2*n+1.

Original entry on oeis.org

1, 1, 2, 4, 1, 2, 8, 8, 16, 1, 2, 12, 16, 48, 32, 64, 1, 2, 16, 24, 96, 96, 256, 128, 256, 1, 2, 20, 32, 160, 192, 640, 512, 1280, 512, 1024, 1, 2, 24, 40, 240, 320, 1280, 1280, 3840, 2560, 6144, 2048, 4096, 1, 2, 28, 48, 336, 480, 2240, 2560, 8960, 7680
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 14 2014

Keywords

Comments

Length of row n = 2*n+1;
T(n,2*n+1) = 4^n = A000302(n);
for n > 0: sum of row n = 7*5^(n-1), cf. A005055.

Links

Crossrefs

Cf. A249095, A000079, A005408 (row lengths), A249308 (central terms), A000302 (right edge), A005055 (row sums, except for initial 1), A013609.
Cf. A013609.

Programs

  • Haskell
    a249307 n k = a249307_tabf !! n !! k
a249307_row n = a249307_tabf !! n
a249307_tabf = map (zipWith (*) a000079_list) a249095_tabf
Showing 1-3 of 3 results.

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