%I #30 Mar 17 2023 07:21:41
%S 0,1,3,4,10,12,28,32,72,80,176,192,416,448,960,1024,2176,2304,4864,
%T 5120,10752,11264,23552,24576,51200,53248,110592,114688,237568,245760,
%U 507904,524288,1081344,1114112,2293760,2359296,4849664,4980736,10223616,10485760,21495808,22020096,45088768,46137344
%N Number of parts in all palindromic compositions of n.
%H Vincenzo Librandi, <a href="/A239632/b239632.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (x + 3*x^2 - 2*x^4)/(1 - 2*x^2)^2.
%F a(n) = Sum_{k=1..n} A051159(n,k)*k.
%F a(n) = 4*a(n-2) - 4*a(n-4) for n > 3. - _Giovanni Resta_, Mar 23 2014
%F a(2k) = (2k+1)*2^(k-1) for k>0, a(2k+1) = (2k+2)*2^(k-1) for k>=0. - _Gregory L. Simay_, Dec 05 2022
%e a(5)=12 because we have: 5, 1+3+1, 2+1+2, 1+1+1+1+1 with a total of 12 parts.
%t nn=30; r=Solve[p==y/(1-x) - y + 1 + y^2*x^2/(1-x^2)*p, p]; CoefficientList[Series[D[p/.r,y]/.y->1, {x,0,nn}], x]
%t CoefficientList[Series[(x + 3 x^2 - 2 x^4)/(1 - 2 x^2)^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 23 2014 *)
%Y Cf. A051159.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Mar 22 2014
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