A036289 a(n) = n*2^n.

0, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, 402653184, 838860800, 1744830464, 3623878656, 7516192768, 15569256448, 32212254720

Offset: 0

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy
• nice

Comments

Right side of the binomial sum Sum_{i = 0..n} (n-2*i)^2 * binomial(n, i) = n*2^n. - Yong Kong (ykong(AT)curagen.com), Dec 28 2000

Let W be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x, y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y, or y is a proper subset of x and there are no z in P(A) such that y is a proper subset of z and z is a proper subset of x. Then a(n) = |W|. - Ross La Haye, Sep 26 2007

Partial sums give A036799. - Vladimir Joseph Stephan Orlovsky, Jul 09 2011

a(n) = n with the bits shifted to the left by n places (new bits on the right hand side are zeros). - Indranil Ghosh, Jan 05 2017

Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - N. J. A. Sloane, Feb 08 2017

Also the circumference of the n-cube connected cycle graph. - Eric W. Weisstein, Sep 03 2017

a(n) is also the number of derangements in S_{n+3} with a descent set of {i, i+1} such that i ranges from 1 to n-2. - Isabella Huang, Mar 17 2018

a(n-1) is also the number of multiplications required to compute the permanent of general n X n matrices using Glynn's formula (see Theorem 2.1 in Glynn). - Stefano Spezia, Oct 27 2021

References

• Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
• A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.29)

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..1000 (first 501 terms from T. D. Noe)
• C. Banderier and S. Schwer, Why Delannoy numbers?, arXiv:math/0411128 [math.CO], 2004.
• David G. Glynn, The permanent of a square matrix, European Journal of Combinatorics, Volume 31, Issue 7, 2010, pp. 1887-1891.
• A. F. Horadam, Oresme numbers, Fib. Quart., 12 (1974), 267-271.
• Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Eric Weisstein's World of Mathematics, Cube-Connected Cycle Graph.
• Eric Weisstein's World of Mathematics, Graph Circumference.
• Index entries for linear recurrences with constant coefficients, signature (4,-4).
• Index entries for sequences related to Benford's law

Crossrefs

Equals 2*A001787. Equals A003261(n) + 1.

Cf. A000079, A036799, A096195, A097064.

Programs

Haskell

a036289 n = n * 2 ^ n
a036289_list = zipWith (*) [0..] a000079_list
-- Reinhard Zumkeller, Mar 05 2012

Maple

g:=1/(1-2*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)*n, n=0..34); # Zerinvary Lajos, Jan 11 2009

Mathematica

Table[n*2^n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2010 *)
LinearRecurrence[{4,-4},{0,2},40] (* Harvey P. Dale, Mar 02 2018 *)

PARI

a(n)=n<Charles R Greathouse IV, Jun 15 2011

Python

a=lambda n: n<Indranil Ghosh, Jan 05 2017

Formula

Main diagonal of array (A085454) defined by T(i, 1) = i, T(1, j) = 2j, T(i, j) = T(i-1, j) + T(i-1, j-1). - Benoit Cloitre, Aug 05 2003

Binomial transform of A005843, the even numbers. - Joshua Zucker, Jan 13 2006

G.f.: 2*x/(1-2*x)^2. - R. J. Mathar, Nov 21 2007

a(n) = A000079(n)*n. - Omar E. Pol, Dec 21 2008

E.g.f.: 2*x exp(2*x). - Geoffrey Critzer, Oct 03 2011

a(n) = A002064(n) - 1. - Reinhard Zumkeller, Mar 16 2013

From Vaclav Kotesovec, Feb 14 2015: (Start)

Sum_{n>=1} 1/a(n) = log(2).

Sum_{n>=1} (-1)^(n+1)/a(n) = log(3/2).

(End)