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A256313 Number of partitions of 3n into exactly 4 parts.

%I A256313 #15 Jul 18 2021 12:30:23
%S A256313 0,0,2,6,15,27,47,72,108,150,206,270,351,441,551,672,816,972,1154,
%T A256313 1350,1575,1815,2087,2376,2700,3042,3422,3822,4263,4725,5231,5760,
%U A256313 6336,6936,7586,8262,8991,9747,10559,11400,12300,13230,14222,15246,16335,17457
%N A256313 Number of partitions of 3n into exactly 4 parts.
%H A256313 Colin Barker, <a href="/A256313/b256313.txt">Table of n, a(n) for n = 0..1000</a>
%H A256313 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,2,-2,0,2,-1).
%F A256313 G.f.: x^2*(x^2+2)*(x^2+x+1) / ((x-1)^4*(x+1)^2*(x^2+1)).
%F A256313 a(n) = (6*n^3+6*n^2-3*n-5+(3*n+1)*(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)+(-1)^((2*n+1-(-1)^n)/4)))/32. - _Luce ETIENNE_, Feb 17 2017
%e A256313 For n=3 the 6 partitions of 3*3 = 9 are [1,1,1,6], [1,1,2,5], [1,1,3,4], [1,2,2,4], [1,2,3,3] and [2,2,2,3].
%t A256313 LinearRecurrence[{2,0,-2,2,-2,0,2,-1},{0,0,2,6,15,27,47,72},60] (* _Harvey P. Dale_, Jul 18 2021 *)
%o A256313 (PARI) concat(0, vector(40, n, k=0; forpart(p=3*n, k++, , [4,4]); k))
%o A256313 (PARI) concat([0,0], Vec(x^2*(x^2+2)*(x^2+x+1)/((x-1)^4*(x+1)^2*(x^2+1)) + O(x^100)))
%Y A256313 Cf. A077043, A256314, A256315.
%K A256313 nonn,easy
%O A256313 0,3
%A A256313 _Colin Barker_, Mar 23 2015