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%I #43 Jan 31 2023 16:08:39
%S 1,1,4,10,24,56,128,288,640,1408,3072,6656,14336,30720,65536,139264,
%T 294912,622592,1310720,2752512,5767168,12058624,25165824,52428800,
%U 109051904,226492416,469762048,973078528,2013265920,4160749568
%N a(0) = a(1) = 1; for n > 1, a(n) = (n+2)*2^(n-2).
%C Binomial transform of A005408 (with interpolated zeros). Binomial transform is A087448. a(n+2) = 2*A045623(n+1); a(n+1) = A001792(n) + (0^n - (-2)^n)/2. The sequence 1,4,10,... given by 2^n*(n+3)/2 - 0^n/2 is the binomial transform of 1,3,3,5,5,...
%C Equals real part of binomial transform of [1, 2*i, 3, 4*i, 5, 6*i, ...]; i=sqrt(-1). - _Gary W. Adamson_, Sep 21 2008
%C An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A057711 (without the leading 0). - _Johannes W. Meijer_, Aug 15 2010
%H Vincenzo Librandi, <a href="/A087447/b087447.txt">Table of n, a(n) for n = 0..3000</a>
%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*(2k+1). - _Paul Barry_, Nov 29 2004
%F From _Colin Barker_, Mar 23 2012: (Start)
%F G.f.: (1-x)*(1-2*x+2*x^2)/(1-2*x)^2.
%F a(n) = 4*a(n-1) - 4*a(n-2) for n > 3. (End)
%F E.g.f.: (1 - x + exp(2*x)*(1 + x))/2. - _Stefano Spezia_, Jan 31 2023
%t Join[{1, 1}, Table[(n + 2) 2^(n - 2), {n, 2, 30}]] (* _Harvey P. Dale_, Feb 22 2011 *)
%Y Essentially same as A079859.
%Y Cf. A001792, A005408, A045623, A057711, A087448, A175655.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Sep 05 2003
%E Definition corrected (by a factor of 2) by _R. J. Mathar_, Feb 21 2009
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