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A256317 Number of partitions of 4n into exactly 6 parts. 2

%I #10 Apr 12 2018 11:55:54

%S 0,0,2,11,35,90,199,391,709,1206,1945,3009,4494,6510,9192,12692,17180,

%T 22856,29941,38677,49342,62239,77695,96079,117788,143247,172929,

%U 207338,247010,292534,344534,403670,470660,546261,631269,726544,832989,951549,1083239

%N Number of partitions of 4n into exactly 6 parts.

%H Colin Barker, <a href="/A256317/b256317.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3,-6,7,-6,6,-6,7,-6,3,-3,3,-1).

%F G.f.: x^2*(x+1)^2*(x^2+1)*(x^4+2*x^3+2*x^2+x+2) / ((x-1)^6*(x^2+x+1)^2*(x^4+x^3+x^2+x+1)).

%e For n=2 the 2 partitions of 4*2 = 8 are [1,1,1,1,1,3] and [1,1,1,1,2,2].

%t Table[Length[IntegerPartitions[4n,{6}]],{n,0,40}] (* or *) LinearRecurrence[ {3,-3,3,-6,7,-6,6,-6,7,-6,3,-3,3,-1},{0,0,2,11,35,90,199,391,709,1206,1945,3009,4494,6510},40] (* _Harvey P. Dale_, Apr 12 2018 *)

%o (PARI) concat(0, vector(40, n, k=0; forpart(p=4*n, k++, , [6,6]); k))

%o (PARI) concat([0,0], Vec(x^2*(x+1)^2*(x^2+1)*(x^4+2*x^3+2*x^2+x+2) / ((x-1)^6*(x^2+x+1)^2*(x^4+x^3+x^2+x+1)) + O(x^100)))

%Y Cf. A238340 (4 parts), A256316 (5 parts).

%K nonn,easy

%O 0,3

%A _Colin Barker_, Mar 23 2015

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