A238706 Sum of the smallest parts of the partitions of 4n into 4 parts with smallest part greater than 1.

0, 2, 11, 36, 89, 183, 335, 565, 894, 1347, 1952, 2738, 3738, 4988, 6525, 8390, 10627, 13281, 16401, 20039, 24248, 29085, 34610, 40884, 47972, 55942, 64863, 74808, 85853, 98075, 111555, 126377, 142626, 160391, 179764, 200838, 223710, 248480, 275249, 304122

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

A. Osorio, A Sequential Allocation Problem: The Asymptotic Distribution of Resources, Munich Personal RePEc Archive, 2014.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

Formula

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

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

Example

Add the numbers > 1 in the last column for a(n):
                                             13 + 1 + 1 + 1
                                             12 + 2 + 1 + 1
                                             11 + 3 + 1 + 1
                                             10 + 4 + 1 + 1
                                              9 + 5 + 1 + 1
                                              8 + 6 + 1 + 1
                                              7 + 7 + 1 + 1
                                             11 + 2 + 2 + 1
                                             10 + 3 + 2 + 1
                                              9 + 4 + 2 + 1
                                              8 + 5 + 2 + 1
                                              7 + 6 + 2 + 1
                                              9 + 3 + 3 + 1
                                              8 + 4 + 3 + 1
                                              7 + 5 + 3 + 1
                                              6 + 6 + 3 + 1
                                              7 + 4 + 4 + 1
                                              6 + 5 + 4 + 1
                                              5 + 5 + 5 + 1
                              9 + 1 + 1 + 1  10 + 2 + 2 + 2
                              8 + 2 + 1 + 1   9 + 3 + 2 + 2
                              7 + 3 + 1 + 1   8 + 4 + 2 + 2
                              6 + 4 + 1 + 1   7 + 5 + 2 + 2
                              5 + 5 + 1 + 1   6 + 6 + 2 + 2
                              7 + 2 + 2 + 1   8 + 3 + 3 + 2
                              6 + 3 + 2 + 1   7 + 4 + 3 + 2
                              5 + 4 + 2 + 1   6 + 5 + 3 + 2
                              5 + 3 + 3 + 1   6 + 4 + 4 + 2
                              4 + 4 + 3 + 1   5 + 5 + 4 + 2
               5 + 1 + 1 + 1  6 + 2 + 2 + 2   7 + 3 + 3 + 3
               4 + 2 + 1 + 1  5 + 3 + 2 + 2   6 + 4 + 3 + 3
               3 + 3 + 1 + 1  4 + 4 + 2 + 2   5 + 5 + 3 + 3
               3 + 2 + 2 + 1  4 + 3 + 3 + 2   5 + 4 + 4 + 3
1 + 1 + 1 + 1  2 + 2 + 2 + 2  3 + 3 + 3 + 3   4 + 4 + 4 + 4
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     0               2              11              36        ..   a(n)

Mathematica

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}]
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 *)
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 *)

Prog

(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
(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

Crossrefs

Cf. A238328, A238340, A238702, A238705.

Sequence in context: A154416 A184538 A316322 * A071244 A005583 A375500

Adjacent sequences: A238703 A238704 A238705 * A238707 A238708 A238709

Keyword

nonn,easy

Author

Wesley Ivan Hurt and Antonio Osorio, Mar 03 2014

Status

approved