This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A238328 #59 Jun 19 2024 22:00:33
%S A238328 4,40,180,544,1280,2592,4732,7968,12636,19120,27808,39168,53716,71960,
%T A238328 94500,121984,155040,194400,240844,295120,358092,430672,513728,608256,
%U A238328 715300,835848,971028,1122016,1289920,1476000,1681564,1907840,2156220,2428144,2724960
%N A238328 Sum of all the parts in the partitions of 4n into 4 parts.
%H A238328 Vincenzo Librandi, <a href="/A238328/b238328.txt">Table of n, a(n) for n = 1..1000</a>
%H A238328 Antonio Osorio, <a href="https://mpra.ub.uni-muenchen.de/56690/">A Sequential Allocation Problem: The Asymptotic Distribution of Resources</a>, Munich Personal RePEc Archive, 2014.
%H A238328 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A238328 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3,-6,6,-3,3,-3,1).
%F A238328 Recurrence: a(1) = 4, with a(n) = (n/(n-1))*a(n-1) + 4n*Sum_{i=0..2n} (floor((4n-2-i)/2)-i) * floor((sign(floor((4n-2-i)/2)-i)+2)/2), n > 1.
%F A238328 G.f.: 4*x*(4*x^6+15*x^5+23*x^4+28*x^3+18*x^2+7*x+1) / ((1-x)^5*(x^2+x+1)^2). - _Colin Barker_, Mar 10 2014
%F A238328 a(n) = 16/9*n^4 + 4/3*n^3 + O(n). - _Ralf Stephan_, May 29 2014
%F A238328 a(n) = 4n*(A238702(n) - A238702(n-1)), n > 1. - _Wesley Ivan Hurt_, May 29 2014
%F A238328 a(n) = 4n * A238340(n). - _Wesley Ivan Hurt_, May 29 2014
%F A238328 E.g.f.: 4*exp(-x/2)*(3*exp(3*x/2)*(8 + x*(37 + x*(27 + 4*x))) + 3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/27. - _Stefano Spezia_, Feb 09 2023
%F A238328 a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - 6*a(n-4) + 6*a(n-5) - 3*a(n-6) + 3*a(n-7) - 3*a(n-8) + a(n-9). - _Wesley Ivan Hurt_, Jun 19 2024
%e A238328 13 + 1 + 1 + 1
%e A238328 12 + 2 + 1 + 1
%e A238328 11 + 3 + 1 + 1
%e A238328 10 + 4 + 1 + 1
%e A238328 9 + 5 + 1 + 1
%e A238328 8 + 6 + 1 + 1
%e A238328 7 + 7 + 1 + 1
%e A238328 11 + 2 + 2 + 1
%e A238328 10 + 3 + 2 + 1
%e A238328 9 + 4 + 2 + 1
%e A238328 8 + 5 + 2 + 1
%e A238328 7 + 6 + 2 + 1
%e A238328 9 + 3 + 3 + 1
%e A238328 8 + 4 + 3 + 1
%e A238328 7 + 5 + 3 + 1
%e A238328 6 + 6 + 3 + 1
%e A238328 7 + 4 + 4 + 1
%e A238328 6 + 5 + 4 + 1
%e A238328 5 + 5 + 5 + 1
%e A238328 9 + 1 + 1 + 1 10 + 2 + 2 + 2
%e A238328 8 + 2 + 1 + 1 9 + 3 + 2 + 2
%e A238328 7 + 3 + 1 + 1 8 + 4 + 2 + 2
%e A238328 6 + 4 + 1 + 1 7 + 5 + 2 + 2
%e A238328 5 + 5 + 1 + 1 6 + 6 + 2 + 2
%e A238328 7 + 2 + 2 + 1 8 + 3 + 3 + 2
%e A238328 6 + 3 + 2 + 1 7 + 4 + 3 + 2
%e A238328 5 + 4 + 2 + 1 6 + 5 + 3 + 2
%e A238328 5 + 3 + 3 + 1 6 + 4 + 4 + 2
%e A238328 4 + 4 + 3 + 1 5 + 5 + 4 + 2
%e A238328 5 + 1 + 1 + 1 6 + 2 + 2 + 2 7 + 3 + 3 + 3
%e A238328 4 + 2 + 1 + 1 5 + 3 + 2 + 2 6 + 4 + 3 + 3
%e A238328 3 + 3 + 1 + 1 4 + 4 + 2 + 2 5 + 5 + 3 + 3
%e A238328 3 + 2 + 2 + 1 4 + 3 + 3 + 2 5 + 4 + 4 + 3
%e A238328 1 + 1 + 1 + 1 2 + 2 + 2 + 2 3 + 3 + 3 + 3 4 + 4 + 4 + 4
%e A238328 4(1) 4(2) 4(3) 4(4) .. 4n
%e A238328 ------------------------------------------------------------------------
%e A238328 4 40 180 544 .. a(n)
%t A238328 CoefficientList[Series[4*(4*x^6 + 15*x^5 + 23*x^4 + 28*x^3 + 18*x^2 + 7*x + 1)/((1 - x)^5*(x^2 + x + 1)^2), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Jun 27 2014 *)
%t A238328 LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {4, 40, 180, 544, 1280, 2592, 4732, 7968, 12636}, 50] (* _Vincenzo Librandi_, Aug 29 2015 *)
%o A238328 (PARI) Vec(-4*x*(4*x^6+15*x^5+23*x^4+28*x^3+18*x^2+7*x+1)/((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ _Colin Barker_, Mar 24 2014
%o A238328 (Magma) I:=[4, 40, 180, 544, 1280, 2592, 4732, 7968, 12636]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+3*Self(n-3)-6*Self(n-4)+6*Self(n-5)-3*Self(n-6)+3*Self(n-7)-3*Self(n-8)+Self(n-9): n in [1..45]]; // _Vincenzo Librandi_, Aug 29 2015
%Y A238328 Cf. A235988, A238340, A238702.
%K A238328 nonn,easy
%O A238328 1,1
%A A238328 _Wesley Ivan Hurt_ and _Antonio Osorio_, Feb 24 2014