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A036289 a(n) = n*2^n. 86
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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

T. D. Noe and 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.

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

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

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

E.g.f.: 2x exp(2x). - 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)

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 *)

PROG

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

(Haskell)

a036289 n = n * 2 ^ n

a036289_list = zipWith (*) [0..] a000079_list

-- Reinhard Zumkeller, Mar 05 2012

(Python) a=lambda n: n<<n # Indranil Ghosh, Jan 05 2017

CROSSREFS

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

Cf. A000079, A036799, A096195, A097064.

Sequence in context: A292218 A097064 A134401 * A294458 A229136 A261452

Adjacent sequences:  A036286 A036287 A036288 * A036290 A036291 A036292

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Dec 11 1999

STATUS

approved

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Last modified June 26 20:30 EDT 2019. Contains 324380 sequences. (Running on oeis4.)