A308775 Sum of all the parts in the partitions of n into 4 parts.

0, 0, 0, 0, 4, 5, 12, 21, 40, 54, 90, 121, 180, 234, 322, 405, 544, 663, 846, 1026, 1280, 1512, 1848, 2162, 2592, 3000, 3536, 4050, 4732, 5365, 6180, 6975, 7968, 8910, 10098, 11235, 12636, 13986, 15618, 17199, 19120, 20951, 23142, 25284, 27808, 30240, 33120

Offset

0,5

Links

Table of n, a(n) for n=0..46.

Index entries for sequences related to partitions

Formula

a(n) = n * A026810(n).

a(n) = A308733(n) + A308758(n) + A308759(n) + A308760(n).

Conjectures from Colin Barker, Jun 24 2019: (Start)

G.f.: x^4*(4 + 5*x + 8*x^2 + 8*x^3 + 10*x^4 + 7*x^5 + 6*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).

a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.

(End)

Example

Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     40          54          90         121         180        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 07 2019

Mathematica

Table[n*Sum[Sum[Sum[1, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

Crossrefs

Cf. A026810, A308733, A308758, A308759, A308760.

Sequence in context: A309479 A352720 A369697 * A305335 A123102 A274077

Adjacent sequences: A308772 A308773 A308774 * A308776 A308777 A308778

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 23 2019

Status

approved