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
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).
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)
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
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Dec 11 1999
STATUS
approved