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A256314 Number of partitions of 3n into exactly 5 parts. 3

%I #11 Jul 22 2019 21:32:11

%S 0,0,1,5,13,30,57,101,164,255,377,540,748,1014,1342,1747,2233,2818,

%T 3507,4319,5260,6351,7599,9027,10642,12470,14518,16814,19366,22204,

%U 25337,28796,32591,36756,41301,46262,51649,57501,63829,70673,78045,85987,94512

%N Number of partitions of 3n into exactly 5 parts.

%H Colin Barker, <a href="/A256314/b256314.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,2,-1,-2,2,1,-2,2,0,-2,1).

%F G.f.: -x^2*(2*x^7+3*x^6+4*x^5+5*x^4+6*x^3+3*x^2+3*x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)).

%e For n=3 the 5 partitions of 3*3 = 9 are [1,1,1,1,5], [1,1,1,2,4], [1,1,1,3,3], [1,1,2,2,3] and [1,2,2,2,2].

%t Table[Length[IntegerPartitions[3n,{5}]],{n,0,50}] (* _Harvey P. Dale_, Jul 21 2019 *)

%o (PARI) concat(0, vector(40, n, k=0; forpart(p=3*n, k++, , [5,5]); k))

%o (PARI) concat([0,0], Vec(-x^2*(2*x^7+3*x^6+4*x^5+5*x^4+6*x^3+3*x^2+3*x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))

%Y Cf. A077043, A256313, A256315.

%K nonn,easy

%O 0,4

%A _Colin Barker_, Mar 23 2015

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