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%I #60 Jun 28 2022 10:59:22
%S 0,2,16,72,256,800,2304,6272,16384,41472,102400,247808,589824,1384448,
%T 3211264,7372800,16777216,37879808,84934656,189267968,419430400,
%U 924844032,2030043136,4437573632,9663676416,20971520000,45365592064,97844723712,210453397504
%N a(n) = 2^n*n^2.
%C "The traveling salesman problem can be solved in time O(n^2 2^n) (where n is the size of the network to visit)." [Wikipedia] - _Jonathan Vos Post_, Apr 10 2006
%C Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - _N. J. A. Sloane_, Feb 08 2017
%D Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
%D Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.
%H Vincenzo Librandi, <a href="/A007758/b007758.txt">Table of n, a(n) for n = 0..1000</a>
%H Konrad Knopp, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM1954.0001.001">Theorie und Anwendung der unendlichen Reihen</a>, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Complexity">Complexity</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>.
%F From _Henry Bottomley_, Jun 13 2001: (Start)
%F a(n) = 2*A014477(n-1).
%F G.f.: 2*x(1+2*x)/(1-2*x)^3.
%F Binomial transform of A002939.
%F Inverse binomial transform of A062189. (End)
%F Sum_{n>=1} 1/a(n) = Pi^2/12 - (1/2)*(log(2))^2. - _Benoit Cloitre_, Apr 05 2002
%F a(n) = Sum_{k=1..n} k*2^k. - _Zerinvary Lajos_, Oct 09 2006
%F E.g.f.: exp(2*x)*(2*x + 4*x^2). - _Geoffrey Critzer_, Aug 28 2013
%F Sum_{n>=1} (-1)^(n+1)/a(n) = -Li_2(-1/2) (A355234). - _Amiram Eldar_, Jun 28 2022
%p seq(seq(k^n*n^k, k=2..2), n=0..25); and seq(2^n*n^2, n=0..25); # _Zerinvary Lajos_, Jul 01 2007
%t Table[n^2 * 2^n, {n, 0, 31}] (* _Alonso del Arte_, Oct 22 2014 *)
%t LinearRecurrence[{6,-12,8},{0,2,16},30] (* _Harvey P. Dale_, Jan 27 2017 *)
%o (Magma) [2^n*n^2: n in [0..30]]; // _Vincenzo Librandi_, Oct 27 2011
%o (PARI) a(n)=n^2<<n \\ _Charles R Greathouse IV_, Oct 28 2014
%Y Cf. A014477, A248917, A355234.
%K nonn,easy
%O 0,2
%A David J. Snook (ua532(AT)freenet.victoria.bc.ca)
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