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%I #52 Dec 12 2022 18:30:33
%S 1,8,36,128,400,1152,3136,8192,20736,51200,123904,294912,692224,
%T 1605632,3686400,8388608,18939904,42467328,94633984,209715200,
%U 462422016,1015021568,2218786816,4831838208,10485760000,22682796032,48922361856,105226698752,225754218496
%N Expansion of (1 + 2*x)/(1 - 2*x)^3.
%C The sequence 0,1,8,... has a(n) = n^2*2^(n-1) and is the binomial transform of the hexagonal numbers A000384 (with leading 0). - _Paul Barry_, Jun 09 2003
%C As 0,1,8,... this is n^2*2^(n-1), the binomial transform of the hexagonal numbers A000384 (include the leading 0). Partial sums are A036826. - _Paul Barry_, Jun 10 2003
%C Sequence gives total value of all possible sums of distinct odd integers with maximum term less than 2n+1. E.g., for a(3) we can have the sums 1, 3, 5, 1+3, 1+5, 3+5, 1+3+5, which sum to 1+3+5+4+6+8+9 = 36. - _Jon Perry_, Feb 06 2004
%C Number of edges on a partially truncated (n+1)-cube (column 2 of A271316).
%H Vincenzo Librandi, <a href="/A014477/b014477.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F O.g.f.: (1 + 2*x)/(1 - 2*x)^3 (see the name).
%F a(n) = (n+1)^2*2^n = A007758(n+1)/2. - _Henry Bottomley_, Jun 13 2001
%F The binomial transform of 0, 1, 8, ... is A077616. - _Paul Barry_, Jul 24 2003
%F a(1)=1, a(n) = 2a(n-1) + (2n-1)*2^(n-1). - _Jon Perry_, Feb 06 2004
%F a(n) = sum of (n+1)-th row of the triangle in A118416. - _Reinhard Zumkeller_, Apr 27 2006
%F a(n) = Sum_{j=0..n} binomial(n,j)*n*j. - _Zerinvary Lajos_, Oct 19 2006
%F E.g.f.: exp(2*x)*(1 + 6*x + 8*x^2/2!). - _Wolfdieter Lang_, Jul 29 2017
%F Sum_{n>=0} 1/a(n) = Pi^2/6 - log(2)^2. - _Daniel Suteu_, Oct 31 2017
%F Sum_{n>=0} (-1)^n/a(n) = -2 * Li_2(-1/2) = -2 * A355234. - _Amiram Eldar_, Oct 01 2022
%p a:=n->sum(binomial(n,j)*n*j,j=0..n): seq(a(n), n=0..25); # _Zerinvary Lajos_, Oct 19 2006
%p a:=n->sum(n*numbcomb(n)/2, j=1..n): seq(a(n), n=1..25); # _Zerinvary Lajos_, Apr 25 2007
%t f[n_]:=(n^2*2^n)/2;Table[f[n],{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 05 2009 *)
%o (Magma) [(n+1)^2*2^n: n in [0..35]]; // _Vincenzo Librandi_, Aug 21 2011
%o (PARI) a(n)=(n+1)^2*2^n \\ _Charles R Greathouse IV_, Apr 07 2016
%Y Cf. A007758, A036826, A118414, A355234.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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