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%I #28 Oct 18 2018 13:26:48
%S 4,32,120,304,600,1056,1708,2560,3672,5080,6776,8832,11284,14112,
%T 17400,21184,25432,30240,35644,41600,48216,55528,63480,72192,81700,
%U 91936,103032,115024,127832,141600,156364,172032,188760,206584,225400,245376,266548,288800
%N Sum of the parts in the partitions of 4n into 4 parts with smallest part = 1.
%C All terms are multiples of 4.
%H Vincenzo Librandi, <a href="/A239056/b239056.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-4,2,-1,2,-1).
%F G.f.: 4*x*(2*x^6+10*x^5+16*x^4+22*x^3+15*x^2+6*x+1) / ((x-1)^4*(x^2+x+1)^2). - _Colin Barker_, Mar 10 2014
%e For a(n) add the parts in the partitions of 4n with smallest part = 1.
%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 + 1 + 1 + 1 9 + 4 + 2 + 1
%e 8 + 2 + 1 + 1 8 + 5 + 2 + 1
%e 7 + 3 + 1 + 1 7 + 6 + 2 + 1
%e 6 + 4 + 1 + 1 9 + 3 + 3 + 1
%e 5 + 5 + 1 + 1 8 + 4 + 3 + 1
%e 7 + 2 + 2 + 1 7 + 5 + 3 + 1
%e 5 + 1 + 1 + 1 6 + 3 + 2 + 1 6 + 6 + 3 + 1
%e 4 + 2 + 1 + 1 5 + 4 + 2 + 1 7 + 4 + 4 + 1
%e 3 + 3 + 1 + 1 5 + 3 + 3 + 1 6 + 5 + 4 + 1
%e 1 + 1 + 1 + 1 3 + 2 + 2 + 1 4 + 4 + 3 + 1 5 + 5 + 5 + 1
%e 4(1) 4(2) 4(3) 4(4) .. 4n
%e ------------------------------------------------------------------------
%e 4 32 120 304 .. a(n)
%t b[n_] := Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; Table[b[n], {n, 50}]
%t LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{4,32,120,304,600,1056,1708,2560},40] (* _Harvey P. Dale_, Oct 18 2018 *)
%o Vec(4*x*(2*x^6+10*x^5+16*x^4+22*x^3+15*x^2+6*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ _Colin Barker_, Sep 22 2014
%Y Cf. A238328, A238340, A238702, A238705, A238706.
%K nonn,easy
%O 1,1
%A _Wesley Ivan Hurt_, Mar 09 2014
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