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A256316 Number of partitions of 4n into exactly 5 parts.

%I A256316 #9 Jun 26 2025 15:34:12
%S A256316 0,0,3,13,37,84,164,291,480,748,1115,1602,2233,3034,4033,5260,6747,
%T A256316 8529,10642,13125,16019,19366,23212,27604,32591,38225,44559,51649,
%U A256316 59553,68331,78045,88759,100540,113456,127578,142979,159733,177918,197613,218899,241860
%N A256316 Number of partitions of 4n into exactly 5 parts.
%H A256316 Colin Barker, <a href="/A256316/b256316.txt">Table of n, a(n) for n = 0..1000</a>
%H A256316 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,4,-4,3,-2,3,-3,1).
%F A256316 G.f.: -x^2*(2*x^6+4*x^5+6*x^4+6*x^3+7*x^2+4*x+3) / ((x-1)^5*(x^2+x+1)*(x^4+x^3+x^2+x+1)).
%F A256316 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 4*a(n-5) - 4*a(n-6) + 3*a(n-7) - 2*a(n-8) + 3*a(n-9) - 3*a(n-10) + a(n-11). - _Wesley Ivan Hurt_, Jun 26 2025
%e A256316 For n=2 the 3 partitions of 4*2 = 8 are [1,1,1,1,4], [1,1,1,2,3] and [1,1,2,2,2].
%o A256316 (PARI) concat(0, vector(40, n, k=0; forpart(p=4*n, k++, , [5,5]); k))
%o A256316 (PARI) concat([0,0], Vec(-x^2*(2*x^6+4*x^5+6*x^4+6*x^3+7*x^2+4*x+3)/((x-1)^5*(x^2+x+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))
%Y A256316 Cf. A238340 (4 parts), A256317 (6 parts).
%K A256316 nonn,easy
%O A256316 0,3
%A A256316 _Colin Barker_, Mar 23 2015