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A239632 Number of parts in all palindromic compositions of n.

%I A239632 #34 Apr 25 2024 11:36:29
%S A239632 0,1,3,4,10,12,28,32,72,80,176,192,416,448,960,1024,2176,2304,4864,
%T A239632 5120,10752,11264,23552,24576,51200,53248,110592,114688,237568,245760,
%U A239632 507904,524288,1081344,1114112,2293760,2359296,4849664,4980736,10223616,10485760,21495808,22020096,45088768,46137344
%N A239632 Number of parts in all palindromic compositions of n.
%H A239632 Vincenzo Librandi, <a href="/A239632/b239632.txt">Table of n, a(n) for n = 0..1000</a>
%H A239632 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-4).
%F A239632 G.f.: (x + 3*x^2 - 2*x^4)/(1 - 2*x^2)^2.
%F A239632 a(n) = Sum_{k=1..n} A051159(n,k)*k.
%F A239632 a(n) = 4*a(n-2) - 4*a(n-4) for n > 3. - _Giovanni Resta_, Mar 23 2014
%F A239632 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
%F A239632 E.g.f.: (2*(1 + x)*cosh(sqrt(2)*x) + sqrt(2)*(1 + 2*x)*sinh(sqrt(2)*x) - 2)/4. - _Stefano Spezia_, Apr 25 2024
%e A239632 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 A239632 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 A239632 CoefficientList[Series[(x + 3 x^2 - 2 x^4)/(1 - 2 x^2)^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 23 2014 *)
%Y A239632 Cf. A051159.
%K A239632 nonn,easy
%O A239632 0,3
%A A239632 _Geoffrey Critzer_, Mar 22 2014