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%I #39 Sep 08 2022 08:46:07
%S 1,10,46,141,334,680,1247,2106,3348,5077,7396,10432,14325,19210,25250,
%T 32621,41490,52056,64531,79114,96040,115557,137896,163328,192137,
%U 224586,260982,301645,346870,397000,452391,513370,580316,653621,733644,820800,915517,1018186,1129258,1249197
%N Sum of the second largest parts of the partitions of 4n into 4 parts.
%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 G.f.: -x*(5*x^6+17*x^5+25*x^4+30*x^3+19*x^2+7*x+1) / ((x-1)^5*(x^2+x+1)^2). - _Colin Barker_, Apr 16 2014
%F Recurrence: Let b(1) = 4, with b(n) = (n/(n-1)) * b(n-1) + 4n*Sum_{i=0..2n} (floor((4n-2-i)/2)-i) * (floor((sign((floor((4n-2-i)/2)-i))+2)/2)) for n>1. Then a(1) = 1, with a(n) = a(n-1) + b(n-1)/(4n-4) + Sum_{j=0..2n} (Sum_{i=j+1..floor((4n-2-j)/2)} i * (floor((sign((floor((4n-2-j)/2)-j))+ 2)/2)) ), for n>1. - _Wesley Ivan Hurt_, Jun 27 2014
%e For a(n) add the numbers in the second columns.
%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 1 10 46 141 .. a(n)
%t CoefficientList[Series[-(5*x^6 + 17*x^5 + 25*x^4 + 30*x^3 + 19*x^2 + 7*x + 1)/((x - 1)^5*(x^2 + x + 1)^2), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Jun 13 2014 *)
%t LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {1, 10, 46, 141, 334, 680, 1247, 2106, 3348}, 50] (* _Vincenzo Librandi_, Aug 29 2015 *)
%o (PARI) Vec(-x*(5*x^6+17*x^5+25*x^4+30*x^3+19*x^2+7*x+1)/((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ _Colin Barker_, Apr 16 2014
%o (Magma) I:=[1,10,46,141,334,680,1247,2106,3348]; [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. A238328, A238340, A238702, A238705, A238706, A239056, A239057, A239059.
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_ and _Antonio Osorio_, Apr 15 2014
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