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%I #35 May 15 2023 12:15:04
%S 1,3,7,17,40,92,208,464,1024,2240,4864,10496,22528,48128,102400,
%T 217088,458752,966656,2031616,4259840,8912896,18612224,38797312,
%U 80740352,167772160,348127232,721420288,1493172224,3087007744,6375342080,13153337344,27111981056
%N Expansion of g.f. (1 - x)^2*(1 + x) / (1 - 2*x)^2.
%C Binomial transform of A029578(n+2). Row sums of number triangle A106471.
%C a(n) is the number of parts equal to 1 or 2 in all the compositions of n + 1. Example: a(2)=7 because in the compositions [3], [1,2], [2,1], and [1,1,1] we have 0 + 2 + 2 + 3 = 7 parts equal to 1 or 2. Equivalently, a(n) = Sum_{k>=0} k*A296559(n+1,k). - _Emeric Deutsch_, Dec 16 2017
%H Colin Barker, <a href="/A106472/b106472.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F a(0)=1, a(1)=3, and a(n) = (3*n + 8)*2^(n-3), for n>=2. [simplified by _Ralf Stephan_, Nov 16 2010]
%F a(n) = 4*a(n-1) - 4*a(n-2) for n > 3. - _Colin Barker_, Dec 16 2017
%F E.g.f.: (exp(2*x)*(4 + 3*x) + x)/4. - _Stefano Spezia_, May 14 2023
%p 1, 3, seq((3*n+8)*2^(n-3), n = 2 .. 27); # _Emeric Deutsch_, Dec 16 2017
%t Join[{1, 3}, LinearRecurrence[{4, -4}, {7, 17}, 30]] (* _Jean-François Alcover_, Dec 16 2017 *)
%o (PARI) x='x+O('x^99); Vec((1+x)*(1-x)^2/(1-2*x)^2) \\ _Altug Alkan_, Dec 16 2017
%Y Cf. A029578, A106471, A296559.
%K easy,nonn
%O 0,2
%A _Paul Barry_, May 03 2005
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