A243011 Sum of the three largest parts in the partitions of 4n into 4 parts.

3, 34, 159, 489, 1161, 2365, 4336, 7323, 11640, 17646, 25702, 36246, 49761, 66720, 87685, 113263, 144039, 180699, 223974, 274561, 333270, 400956, 478428, 566620, 666511, 779022, 905211, 1046181, 1202965, 1376745, 1568748, 1780119, 2012164, 2266234, 2543586

Offset

1,1

Links

Table of n, a(n) for n=1..35.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-6,6,-3,3,-3,1).

Formula

a(n) = A238328(n) - A238702(n).

a(n) = A239667(n) + A241084(n) + A242727(n).

a(n) = 4n * A238340(n) - Sum_{i=1..n} A238340(i).

a(n) = (4n-1) * A238702(n) - 4n * A238702(n-1), n > 1.

a(n) = A238328(n) - (1/4) * Sum_{i=1..n} A238328(i)/i.

G.f.: -x*(16*x^6+58*x^5+87*x^4+105*x^3+66*x^2+25*x+3) / ((x-1)^5*(x^2+x+1)^2). - Colin Barker, Sep 22 2014

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - 6*a(n-4) + 6*a(n-5) - 3*a(n-6) + 3*a(n-7) - 3*a(n-8) + a(n-9). - Wesley Ivan Hurt, Jun 20 2024

Example

Add up the numbers in the first three columns 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
------------------------------------------------------------------------
     3               34            159             489        ..   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}]; Table[a[n] - Sum[a[i]/i, {i, n}]/4, {n, 30}]

Prog

(PARI) Vec(-x*(16*x^6+58*x^5+87*x^4+105*x^3+66*x^2+25*x+3)/((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

Crossrefs

Cf. A238328, A238340, A238702, A239667, A241084.

Sequence in context: A274871 A042041 A180775 * A279130 A266696 A141789

Adjacent sequences: A243008 A243009 A243010 * A243012 A243013 A243014

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 28 2014

Status

approved