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A308760
Sum of the largest parts of the partitions of n into 4 parts.
4
0, 0, 0, 0, 1, 2, 5, 9, 17, 25, 41, 57, 84, 112, 154, 197, 262, 325, 414, 506, 629, 751, 915, 1078, 1289, 1501, 1767, 2034, 2370, 2701, 3108, 3519, 4014, 4506, 5100, 5691, 6393, 7095, 7917, 8739, 9703, 10658, 11765, 12876, 14150, 15418, 16874, 18324, 19974
OFFSET
0,6
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (n-i-j-k).
a(n) = A308775(n) - A308733(n) - A308758(n) - A308759(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4*(1 + 2*x + 4*x^2 + 5*x^3 + 6*x^4 + 4*x^5 + 3*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
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n | 8 9 10 11 12 ...
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a(n) | 17 25 41 57 84 ...
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- Wesley Ivan Hurt, Sep 07 2019
MATHEMATICA
Table[Sum[Sum[Sum[n - i - j - k, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 22 2019
STATUS
approved