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%I #15 Jul 28 2022 21:51:37
%S 0,0,0,1,2,5,11,20,35,58,90,136,199,282,391,532,709,931,1206,1540,
%T 1945,2432,3009,3692,4494,5427,6510,7760,9192,10829,12692,14800,17180,
%U 19858,22856,26207,29941,34085,38677,43752,49342,55491,62239,69624,77695,86499
%N Number of partitions of 2n into exactly 6 parts.
%C The number of partitions of 2*(n-3) into at most 6 parts. - _Colin Barker_, Mar 31 2015
%H Colin Barker, <a href="/A256310/b256310.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,-3,1,1,0,0,1,1,-3,0,0,2,-1).
%F G.f.: x^3*(x^4+x^3+x^2+1) / ((x-1)^6*(x+1)*(x^2+x+1)^2*(x^4+x^3+x^2+x+1)).
%F a(n) = 2*a(n-1) - 3*a(n-4) + a(n-5) + a(n-6) + a(n-9) + a(n-10) - 3*a(n-11) + 2*a(n-14) - a(n-15). - _Wesley Ivan Hurt_, Jul 28 2022
%e For n=4 the 2 partitions of 2*4 = 8 are [1,1,1,1,1,3] and [1,1,1,1,2,2].
%t CoefficientList[Series[x^3 (x^4 + x^3 + x^2 + 1) / ((x - 1)^6 (x + 1) (x^2 + x + 1)^2 (x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 22 2015 *)
%o (PARI) concat(0, vector(40, n, k=0; forpart(p=2*n, k++, , [6,6]); k))
%o (PARI) concat([0,0,0], Vec(x^3*(x^4+x^3+x^2+1)/((x-1)^6*(x+1)*(x^2+x+1)^2*(x^4+x^3+x^2+x+1)) + O(x^100)))
%Y Cf. Similar sequences: A000212 (3 parts), A001477 (2 parts), A014126 (4 parts), A256309 (5 parts).
%K nonn,easy
%O 0,5
%A _Colin Barker_, Mar 22 2015
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