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 1-2 of 2 results.

A097801 a(n) = (2*n)!/(n!*2^(n-1)).

Original entry on oeis.org

2, 2, 6, 30, 210, 1890, 20790, 270270, 4054050, 68918850, 1309458150, 27498621150, 632468286450, 15811707161250, 426916093353750, 12380566707258750, 383797567925021250, 12665319741525701250, 443286190953399543750, 16401589065275783118750, 639661973545755541631250
Offset: 0

Views

Author

N. J. A. Sloane, Sep 21 2004

Keywords

Comments

Right-hand edge of triangle in A097749.
Also, the number of ways to paint the 2*n cells of dimension n - 1 that bound a regular convex n-cube polytope using exactly 2n colors where n > 0 is the dimension of Euclidean space. - Frank M Jackson, Aug 13 2018

Links

Crossrefs

Cf. A001147, A097749.

Programs

  • GAP
    List([0..20],n->Factorial(2*n)/(Factorial(n)*2^(n-1))); # Muniru A Asiru, Aug 21 2018
  • Magma
    [Factorial(2*n)/(Factorial(n)*2^(n-1)): n in [0..20]]; // Vincenzo Librandi, Aug 21 2018
  • Maple
    a:= proc(n) a(n):= `if`(n=0, 2, a(n-1)*(2*n-1)) end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 27 2013
  • Mathematica
    FoldList[Times, 2, Range[1, 39, 2]] (* Arkadiusz Wesolowski, May 08 2012 *)
2(2*Range[0,20]-1)!! (* Harvey P. Dale, Apr 22 2013 *)
CoefficientList[Series[2/Sqrt[1-2*x], {x, 0, 45}], x]*Table[k !, {k, 0, 45}] (* Stefano Spezia, Sep 04 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(2*(1-2*x)^(-1/2))) \\ Altug Alkan, Sep 05 2018

Formula

a(n) = 2*A001147(n).
a(n) = 2*(2*n-1)!!. - Johannes W. Meijer, Nov 12 2009
E.g.f.: 2/sqrt(1 - 2*x). - Sergei N. Gladkovskii, Jul 06 2012
G.f.: G(0), where G(k)= 1 + 1/(1 - x*(2*k + 1)/(x*(2*k + 1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = 2 * Product_{i=1..n} denominator(i!/(2*i - 1)). - Wesley Ivan Hurt, Oct 12 2013
D-finite with recurrence: a(n) + (-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
From Amiram Eldar, Feb 24 2021: (Start)
Sum_{n>=0} 1/a(n) = 1/2 + sqrt(e*Pi/2)*erf(1/sqrt(2))/2, where erf(x) is the error function.
Sum_{n>=0} (-1)^n/a(n) = 1/2 - sqrt(Pi/(2*e))*erfi(1/sqrt(2))/2, where erfi(x) is the imaginary error function. (End)

A097474 Triangle T(n,k) read by rows: see formula lines for definition.

Original entry on oeis.org

1, -1, 6, 2, -10, 10, -17, 84, -70, 28, 124, -612, 504, -168, 36, -2764, 13640, -11220, 3696, -660, 88, 43688, -215592, 177320, -58344, 10296, -1144, 104, -1859138, 9174480, -7545720, 2482480, -437580, 48048, -3640, 240, 51236656, -252842768, 207954880, -68414528, 12057760, -1322464, 99008, -5440
Offset: 0

Views

Author

N. J. A. Sloane, Sep 21 2004

Keywords

Examples

			Triangle begins:
1
-1 6
2 -10 10
-17 84 -70 28
124 -612 504 -168 36

References

  • H. W. Gould, Power sum identities for arbitrary symmetric arrays, SIAM J. Appl. Math., 17 (1969), 307-316.

Crossrefs

Cf. A097578, A097716, A097749.

Programs

  • Mathematica
    T[n_, n_] := (2n + 1)2^Floor[(n + 1)/2]; T[n_, k_] := (Binomial[2n + 1, 2k] - Sum[ Binomial[2n + 1, 2j + 1]*2^-Floor[(j + 3)/2]*T[j, k], {j, k, n - 1}])(2^Floor[(n + 1)/2]); Flatten[ Table[ T[n, k], {n, 0, 8}, {k, 0, n}]] (* Robert G. Wilson v, May 10 2005 *)

Formula

T(n, n) = (2n+1)2^floor((n+1)/2), n >= 0.
2^-floor((n+2)/2)*T(n+1, k) = binomial(2n+3, 2k) - Sum_{j=k..n} binomial(2n+3, 2j+1)*2^-floor((j+3)/2)*T(j, k), k=0..n.

Extensions

More terms from Emeric Deutsch, Dec 24 2004
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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