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%I #119 Nov 03 2023 10:24:39
%S 0,2,8,24,64,160,384,896,2048,4608,10240,22528,49152,106496,229376,
%T 491520,1048576,2228224,4718592,9961472,20971520,44040192,92274688,
%U 192937984,402653184,838860800,1744830464,3623878656,7516192768,15569256448,32212254720
%N a(n) = n*2^n.
%C 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
%C 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
%C Partial sums give A036799. - _Vladimir Joseph Stephan Orlovsky_, Jul 09 2011
%C 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
%C Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - _N. J. A. Sloane_, Feb 08 2017
%C Also the circumference of the n-cube connected cycle graph. - _Eric W. Weisstein_, Sep 03 2017
%C 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
%C 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
%D Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
%D 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)
%H Indranil Ghosh, <a href="/A036289/b036289.txt">Table of n, a(n) for n = 0..1000</a> (first 501 terms from T. D. Noe)
%H C. Banderier and S. Schwer, <a href="http://arXiv.org/abs/math.CO/0411128">Why Delannoy numbers?</a>, arXiv:math/0411128 [math.CO], 2004.
%H David G. Glynn, <a href="https://doi.org/10.1016/j.ejc.2010.01.010">The permanent of a square matrix</a>, European Journal of Combinatorics, Volume 31, Issue 7, 2010, pp. 1887-1891.
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/12-3/horadam.pdf">Oresme numbers</a>, Fib. Quart., 12 (1974), 267-271.
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cube-ConnectedCycleGraph.html">Cube-Connected Cycle Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCircumference.html">Graph Circumference</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F 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
%F Binomial transform of A005843, the even numbers. - _Joshua Zucker_, Jan 13 2006
%F G.f.: 2*x/(1-2*x)^2. - _R. J. Mathar_, Nov 21 2007
%F a(n) = A000079(n)*n. - _Omar E. Pol_, Dec 21 2008
%F E.g.f.: 2*x exp(2*x). - _Geoffrey Critzer_, Oct 03 2011
%F a(n) = A002064(n) - 1. - _Reinhard Zumkeller_, Mar 16 2013
%F From _Vaclav Kotesovec_, Feb 14 2015: (Start)
%F Sum_{n>=1} 1/a(n) = log(2).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(3/2).
%F (End)
%p 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
%t Table[n*2^n, {n, 0, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 18 2010 *)
%t LinearRecurrence[{4,-4},{0,2},40] (* _Harvey P. Dale_, Mar 02 2018 *)
%o (PARI) a(n)=n<<n \\ _Charles R Greathouse IV_, Jun 15 2011
%o (Haskell)
%o a036289 n = n * 2 ^ n
%o a036289_list = zipWith (*) [0..] a000079_list
%o -- _Reinhard Zumkeller_, Mar 05 2012
%o (Python) a=lambda n: n<<n # _Indranil Ghosh_, Jan 05 2017
%Y Equals 2*A001787. Equals A003261(n) + 1.
%Y Cf. A000079, A036799, A096195, A097064.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, Dec 11 1999
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