A256328 - OEIS

%I #7 Mar 22 2020 13:52:53

%S 0,2,15,47,108,206,351,551,816,1154,1575,2087,2700,3422,4263,5231,

%T 6336,7586,8991,10559,12300,14222,16335,18647,21168,23906,26871,30071,

%U 33516,37214,41175,45407,49920,54722,59823,65231,70956,77006,83391,90119,97200

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

%H Colin Barker, <a href="/A256328/b256328.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).

%F a(n) = (-1+(-1)^n+6*n^2+12*n^3)/8.

%F a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4.

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

%e For n=1 the 2 partitions of 6*1 = 6 are [1,1,1,3] and [1,1,2,2].

%t LinearRecurrence[{3,-2,-2,3,-1},{0,2,15,47,108},50] (* _Harvey P. Dale_, Mar 22 2020 *)

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

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

%Y Cf. A256327 (5n), A256329 (7n).

%K nonn,easy

%O 0,2

%A _Colin Barker_, Mar 25 2015