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A256321 Number of partitions of 5n into exactly 3 parts.

%I A256321 #10 Oct 29 2017 12:48:07
%S A256321 0,2,8,19,33,52,75,102,133,169,208,252,300,352,408,469,533,602,675,
%T A256321 752,833,919,1008,1102,1200,1302,1408,1519,1633,1752,1875,2002,2133,
%U A256321 2269,2408,2552,2700,2852,3008,3169,3333,3502,3675,3852,4033,4219,4408,4602
%N A256321 Number of partitions of 5n into exactly 3 parts.
%H A256321 Colin Barker, <a href="/A256321/b256321.txt">Table of n, a(n) for n = 0..1000</a>
%H A256321 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A256321 a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6) for n>5.
%F A256321 G.f.: -x*(x^2+2*x+2)*(2*x^2+2*x+1) / ((x-1)^3*(x+1)*(x^2+x+1)).
%e A256321 For n=1 the 2 partitions of 5*1 = 5 are [1, 1, 3] and [1, 2, 2].
%t A256321 Length /@ (Total /@ IntegerPartitions[5 #, {3}] & /@ Range[0, 47]) (* _Michael De Vlieger_, Mar 24 2015 *)
%t A256321 LinearRecurrence[{1,1,0,-1,-1,1},{0,2,8,19,33,52},50] (* _Harvey P. Dale_, Oct 29 2017 *)
%o A256321 (PARI) concat(0, vector(40, n, k=0; forpart(p=5*n, k++, , [3,3]); k))
%o A256321 (PARI) concat(0, Vec(-x*(x^2+2*x+2)*(2*x^2+2*x+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^100)))
%Y A256321 Cf. A033428 (6n), A256320 (4n), A256322 (7n).
%K A256321 nonn,easy
%O A256321 0,2
%A A256321 _Colin Barker_, Mar 24 2015