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%I #15 Mar 06 2023 08:24:54
%S 0,0,0,1,3,7,13,23,37,57,84,119,164,221,291,377,480,603,748,918,1115,
%T 1342,1602,1898,2233,2611,3034,3507,4033,4616,5260,5969,6747,7599,
%U 8529,9542,10642,11835,13125,14518,16019,17633,19366,21224,23212,25337,27604
%N Number of partitions of 2n into exactly 5 parts.
%H Colin Barker, <a href="/A256309/b256309.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,1,0,-1,1,1,0,-2,1).
%F G.f.: -x^3*(x^4+x^2+x+1) / ((x-1)^5*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)).
%e For n=4 the 3 partitions of 2*4 = 8 are [1,1,1,1,4], [1,1,1,2,3] and [1,1,2,2,2].
%t CoefficientList[Series[- x^3 (x^4 + x^2 + x + 1) / ((x - 1)^5 (x + 1) (x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 22 2015 *)
%t LinearRecurrence[{2,0,-1,-1,1,0,-1,1,1,0,-2,1},{0,0,0,1,3,7,13,23,37,57,84,119},50] (* _Harvey P. Dale_, Mar 06 2023 *)
%o (PARI) concat(0, vector(40, n, k=0; forpart(p=2*n, k++, , [5,5]); k))
%o (PARI) concat([0,0,0], Vec(-x^3*(x^4+x^2+x+1)/((x-1)^5*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))
%Y Cf. Similar sequences: A000212 (3 parts), A001477 (2 parts), A014126 (4 parts), A256310 (6 parts).
%K nonn,easy
%O 0,5
%A _Colin Barker_, Mar 22 2015
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