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.

A193132 a(n) = 3n*4^(2n-1).

Original entry on oeis.org

12, 384, 9216, 196608, 3932160, 75497472, 1409286144, 25769803776, 463856467968, 8246337208320, 145135534866432, 2533274790395904, 43910096366862336, 756604737398243328, 12970366926827028480, 221360928884514619392, 3763135791036748529664
Offset: 1

Views

Author

Eric W. Weisstein, Jul 16 2011

Keywords

Comments

a(n) is the number of spanning trees for the 2n-crossed prism graph with n >= 2.
Sequence extended to n=1 using the closed form.

Links

Crossrefs

Cf. A267796.

Programs

Formula

O.g.f.: 12*x/(16*x-1)^2.
a(n) = 32*a(n-1) - 256*a(n-2). - Harvey P. Dale, Apr 08 2015
From Amiram Eldar, Apr 17 2022: (Start)
a(n) = 3*A267796(n-1).
Sum_{n>=1} 1/a(n) = (4/3)*log(16/15).
Sum_{n>=1} (-1)^(n+1)/a(n) = (4/3)*log(17/16). (End)

A239739 a(n) = n*4^(2*n+1).

Original entry on oeis.org

0, 64, 2048, 49152, 1048576, 20971520, 402653184, 7516192768, 137438953472, 2473901162496, 43980465111040, 774056185954304, 13510798882111488, 234187180623265792, 4035225266123964416, 69175290276410818560, 1180591620717411303424, 20070057552195992158208
Offset: 0

Views

Author

Peter Luschny, Mar 26 2014

Keywords

Comments

Appears in asymptotic expansions of the logarithm of the central binomial and the Catalan numbers. (See Kessler and Schiff, page 2.)

Links

Crossrefs

Cf. A219931, A220002, A220422, A267796, A013709.

Programs

  • Magma
    [n*4^(2*n+1): n in [0..25]]; // Vincenzo Librandi, Apr 25 2014
  • Mathematica
    CoefficientList[Series[64 x /(1 - 16 x)^2, {x, 0, 20}], x] (* Vincenzo Librandi, Apr 25 2014 *)
LinearRecurrence[{32,-256},{0,64},20] (* Harvey P. Dale, May 06 2021 *)

Formula

G.f.: 64*x / (1 - 16*x)^2. [Bruno Berselli, Mar 26 2014]
(n-1)*a(n) - 16*n*a(n-1) = 0. [Bruno Berselli, Mar 26 2014]
a(n) = n*A013709(n). - Michel Marcus, Jan 30 2016

A267982 a(n) = 4*n*Catalan(n)^2.

Original entry on oeis.org

0, 4, 32, 300, 3136, 35280, 418176, 5153148, 65436800, 851005584, 11284224640, 152054927024, 2076911622912, 28698821320000, 400547241561600, 5639401174441500, 80010548981049600, 1142928467041798800, 16425988397113680000, 237364657887402183600
Offset: 0

Views

Author

Ralf Steiner, Jan 23 2016

Keywords

Comments

The series whose terms are the quotients a(n)/A013709(n) (modified (4n+0) Wallis-Lambert-series-1) is convergent to 4*(1-3/Pi). Proof: Both the Wallis-Lambert-series-1=4/Pi-1 and the elliptic Euler-series=1-2/Pi are absolutely convergent series. Thus any linear combination of the terms of these series will be also absolutely convergent to the value of the linear combination of these series - in this case to 4(1-3/Pi). Q.E.D.

Examples

			For n=3, a(3)=300.

Links

Crossrefs

Cf. A013709, A267796, A268085.

Programs

  • Magma
    [Catalan(n)^2*4*n: n in [0..20]]; // Vincenzo Librandi, Jan 24 2016
  • Mathematica
    Table[CatalanNumber[n]^2 (4 n + 0), {n, 0, 20}]
  • PARI
    a(n) = 4*n*(binomial(2*n, n)/(n+1))^2; \\ Michel Marcus, Jan 24 2016
  • Python
    from _future_ import division
A267982_list, b = [0], 4
for n in range(1,10**2):
    A267982_list.append(b)
    b = b*4*(n+1)*(2*n+1)**2//(n*(n+2)**2) # Chai Wah Wu, Jan 28 2016

Formula

a(n) = 4*A268085(n).
a(n+1) = a(n)*4*(n+1)*(2*n+1)^2/(n*(n+2)^2) for n > 0. - Chai Wah Wu, Jan 28 2016

Extensions

More terms from Vincenzo Librandi, Jan 24 2016
Showing 1-3 of 3 results.

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

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