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A256239 Sum of all the parts in the partitions of 6n into 6 parts.

%I A256239 #16 Mar 07 2025 17:08:59
%S A256239 0,6,132,1044,4776,15960,43416,102144,215712,419040,761520,1310628,
%T A256239 2155752,3412656,5228076,7784910,11307648,16068264,22392504,30666570,
%U A256239 41344080,54953640,72106452,93504798,119950416,152353650,191742720,239273514,296239776,364083690
%N A256239 Sum of all the parts in the partitions of 6n into 6 parts.
%H A256239 Colin Barker, <a href="/A256239/b256239.txt">Table of n, a(n) for n = 0..1000</a>
%H A256239 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,3,-9,3,10,-10,-3,9,-3,-5,5,1,-3,1).
%F A256239 a(n) = 6*n*A256226(n).
%F A256239 G.f.: -6*x*(9*x^13 +77*x^12 +247*x^11 +485*x^10 +744*x^9 +990*x^8 +1109*x^7 +1029*x^6 +809*x^5 +551*x^4 +301*x^3 +109*x^2 +19*x +1) / ((x -1)^7*(x +1)^2*(x^4 +x^3 +x^2 +x +1)^2).
%e A256239 For n=2 there are 11 partitions of 6*2 = 12, so a(2) = 11*12 = 132.
%t A256239 Plus @@ Total /@ IntegerPartitions[6 #, {6}] & /@ Range[0, 29] (* _Michael De Vlieger_, Mar 20 2015 *)
%t A256239 CoefficientList[Series[- 6 x (9 x^13 + 77 x^12 + 247 x^11 + 485 x^10 + 744 x^9 + 990 x^8 + 1109 x^7 + 1029 x^6 + 809 x^5 + 551 x^4 + 301 x^3 + 109 x^2 + 19 x + 1) / ((x - 1)^7 (x + 1)^2 (x^4 + x^3 + x^2 + x + 1)^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 20 2015 *)
%t A256239 LinearRecurrence[{3,-1,-5,5,3,-9,3,10,-10,-3,9,-3,-5,5,1,-3,1},{0,6,132,1044,4776,15960,43416,102144,215712,419040,761520,1310628,2155752,3412656,5228076,7784910,11307648},30] (* _Harvey P. Dale_, Mar 07 2025 *)
%o A256239 (PARI)
%o A256239 concat(0, Vec(-6*x*(9*x^13 +77*x^12 +247*x^11 +485*x^10 +744*x^9 +990*x^8 +1109*x^7 +1029*x^6 +809*x^5 +551*x^4 +301*x^3 +109*x^2 +19*x +1) / ((x -1)^7*(x +1)^2*(x^4 +x^3 +x^2 +x +1)^2) + O(x^100)))
%Y A256239 Cf. A235988, A238328, A256225, A256235.
%K A256239 nonn,easy
%O A256239 0,2
%A A256239 _Colin Barker_, Mar 20 2015