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.

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A056616 Numerator of binomial(2n,n)/(2n+1).

Original entry on oeis.org

1, 2, 6, 20, 70, 252, 924, 1144, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 200360130, 466721244, 9075135300, 11781754600, 137846528820, 538257874440, 140273264248, 8233430727600, 32247603683100
Offset: 0

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Author

N. J. A. Sloane, Aug 28 2000

Keywords

Comments

The denominators are given in A056617.
It is easy to type binomial(2n,n)/(2n+1) by mistake, when one really wants the Catalan numbers binomial(2n,n)/(n+1), A000108.
Differs from A000984 at positions in A101681.

Examples

			The rationals r(n) begin: 1, 2/3, 6/5, 20/7, 70/9, 252/11, 924/13, 1144/5, 12870/17, ...

Links

Crossrefs

Cf. A000984, A056617, A000108, A101681.

Programs

  • Maple
    seq(numer(binomial(2*n,n)/(2*n+1)), n=0..50); # Robert Israel, Feb 21 2017
  • Mathematica
    Numerator[Table[Binomial[2n,n]/(2n+1),{n,0,30}]] (* Harvey P. Dale, Jul 25 2013 *)
  • PARI
    a(n) = numerator(binomial(2*n, n)/(2*n+1)) \\ Felix Fröhlich, Feb 21 2017

Formula

Numerators of the rationals r(n) = binomial(2n,n)/(2n+1) with G.f.: 1/(2*sqrt(x))*arcsin(2*sqrt(x)). [Vladimir Kruchinin, May 31 2013]

A073076 Numbers k such that 2*k+1 divides C(2*k,k).

Original entry on oeis.org

97, 136, 178, 192, 199, 292, 313, 332, 448, 467, 472, 478, 487, 535, 542, 577, 604, 617, 697, 773, 790, 797, 852, 885, 940, 962, 967, 997, 1017, 1045, 1096, 1127, 1147, 1165, 1182, 1202, 1237, 1291, 1292, 1319, 1332, 1339, 1345, 1354, 1368, 1397, 1414
Offset: 1

Views

Author

Benoit Cloitre, Aug 17 2002

Keywords

Comments

Integers k such that A056617(k) = 1. - Michel Marcus, May 27 2019
Numbers n such that A005408(n) divides A000984(n). - Felix Fröhlich, May 27 2019

Links

Crossrefs

Cf. A000984, A005408, A056616, A056617, A080394.

Programs

  • Mathematica
    Select[Range@ 1500, Mod[Binomial[2 #, #], 2 # + 1] == 0 &] (* Michael De Vlieger, May 27 2019 *)
  • PARI
    isok(n) = ! (binomial(2*n, n) % (2*n+1)); \\ Michel Marcus, Nov 28 2013
Showing 1-2 of 2 results.

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

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