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 A238328 Sum of all the parts in the partitions of 4n into 4 parts. 19

%I

%S 4,40,180,544,1280,2592,4732,7968,12636,19120,27808,39168,53716,71960,

%T 94500,121984,155040,194400,240844,295120,358092,430672,513728,608256,

%U 715300,835848,971028,1122016,1289920,1476000,1681564,1907840,2156220,2428144,2724960

%N Sum of all the parts in the partitions of 4n into 4 parts.

%H Vincenzo Librandi, <a href="/A238328/b238328.txt">Table of n, a(n) for n = 1..1000</a>

%H A. Osorio, <a href="http://mpra.ub.uni-muenchen.de/56690/1/MPRA_paper_56690.pdf">A Sequential Allocation Problem: The Asymptotic Distribution of Resources</a>, Munich Personal RePEc Archive, 2014.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <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 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 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 a(n) = 16/9*n^4 + 4/3*n^3 + O(n). - _Ralf Stephan_, May 29 2014

%F a(n) = 4n*(A238702(n) - A238702(n-1)), n > 1. - _Wesley Ivan Hurt_, May 29 2014

%F a(n) = 4n * A238340(n). - _Wesley Ivan Hurt_, May 29 2014

%e 13 + 1 + 1 + 1

%e 12 + 2 + 1 + 1

%e 11 + 3 + 1 + 1

%e 10 + 4 + 1 + 1

%e 9 + 5 + 1 + 1

%e 8 + 6 + 1 + 1

%e 7 + 7 + 1 + 1

%e 11 + 2 + 2 + 1

%e 10 + 3 + 2 + 1

%e 9 + 4 + 2 + 1

%e 8 + 5 + 2 + 1

%e 7 + 6 + 2 + 1

%e 9 + 3 + 3 + 1

%e 8 + 4 + 3 + 1

%e 7 + 5 + 3 + 1

%e 6 + 6 + 3 + 1

%e 7 + 4 + 4 + 1

%e 6 + 5 + 4 + 1

%e 5 + 5 + 5 + 1

%e 9 + 1 + 1 + 1 10 + 2 + 2 + 2

%e 8 + 2 + 1 + 1 9 + 3 + 2 + 2

%e 7 + 3 + 1 + 1 8 + 4 + 2 + 2

%e 6 + 4 + 1 + 1 7 + 5 + 2 + 2

%e 5 + 5 + 1 + 1 6 + 6 + 2 + 2

%e 7 + 2 + 2 + 1 8 + 3 + 3 + 2

%e 6 + 3 + 2 + 1 7 + 4 + 3 + 2

%e 5 + 4 + 2 + 1 6 + 5 + 3 + 2

%e 5 + 3 + 3 + 1 6 + 4 + 4 + 2

%e 4 + 4 + 3 + 1 5 + 5 + 4 + 2

%e 5 + 1 + 1 + 1 6 + 2 + 2 + 2 7 + 3 + 3 + 3

%e 4 + 2 + 1 + 1 5 + 3 + 2 + 2 6 + 4 + 3 + 3

%e 3 + 3 + 1 + 1 4 + 4 + 2 + 2 5 + 5 + 3 + 3

%e 3 + 2 + 2 + 1 4 + 3 + 3 + 2 5 + 4 + 4 + 3

%e 1 + 1 + 1 + 1 2 + 2 + 2 + 2 3 + 3 + 3 + 3 4 + 4 + 4 + 4

%e 4(1) 4(2) 4(3) 4(4) .. 4n

%e ------------------------------------------------------------------------

%e 4 40 180 544 .. a(n)

%t 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 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 (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 (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 Cf. A235988, A238340.

%K nonn,easy

%O 1,1

%A _Wesley Ivan Hurt_ and _Antonio Osorio_, Feb 24 2014

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Last modified July 21 02:02 EDT 2019. Contains 325189 sequences. (Running on oeis4.)