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A308775
Sum of all the parts in the partitions of n into 4 parts.
4
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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 23 2019
STATUS
approved