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Sum of the smallest parts of the partitions of 4n into 4 parts with smallest part greater than 1.
10

%I #35 Jan 06 2023 19:44:07

%S 0,2,11,36,89,183,335,565,894,1347,1952,2738,3738,4988,6525,8390,

%T 10627,13281,16401,20039,24248,29085,34610,40884,47972,55942,64863,

%U 74808,85853,98075,111555,126377,142626,160391,179764,200838,223710,248480,275249,304122

%N Sum of the smallest parts of the partitions of 4n into 4 parts with smallest part greater than 1.

%H Vincenzo Librandi, <a href="/A238706/b238706.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_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-5,6,-4,1).

%F G.f.: x^2*(x-2)*(x+1)*(2*x^2+x+1) / ((x-1)^5*(x^2+x+1)). - _Colin Barker_, Mar 23 2014

%F a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 5*a(n-4) + 6*a(n-5) - 4*a(n-6) + a(n-7) for n > 7. - _Wesley Ivan Hurt_, Oct 07 2017

%e Add the numbers > 1 in the last column for a(n):

%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 0 2 11 36 .. a(n)

%t a[1] = 4; a[n_] := (n/(n - 1))*a[n - 1] + 4 n*Sum[(Floor[(4 n - 2 - i)/2] - i)*(Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; b[n_] := a[n]/(4 n); b[0] = 0; c[1] = 1; c[n_] := b[n] + c[n - 1]; Table[c[n] - (b[n] - b[n - 1]), {n, 50}]

%t CoefficientList[Series[x (x - 2) (x + 1) (2 x^2 + x + 1)/((x - 1)^5 (x^2 + x + 1)), {x, 0, 40}],x] (* _Vincenzo Librandi_, Mar 24 2014 *)

%t Table[Total[Select[IntegerPartitions[4n,{4}],#[[-1]]>1&][[All,-1]]],{n,40}] (* or *) LinearRecurrence[{4,-6,5,-5,6,-4,1},{0,2,11,36,89,183,335},40] (* _Harvey P. Dale_, Jan 06 2023 *)

%o (PARI) concat(0, Vec(x^2*(x-2)*(x+1)*(2*x^2+x+1)/((x-1)^5*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, Mar 23 2014

%o (Magma) I:=[0,2,11,36,89,183,335]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+5*Self(n-3)-5*Self(n-4)+6*Self(n-5)-4*Self(n-6)+Self(n-7): n in [1..40]]; // _Vincenzo Librandi_, Mar 24 2014

%Y Cf. A238328, A238340, A238702, A238705.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_ and _Antonio Osorio_, Mar 03 2014