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A308758
Sum of the third largest parts of the partitions of n into 4 parts.
4
0, 0, 0, 0, 1, 1, 2, 4, 7, 9, 15, 20, 29, 38, 51, 64, 86, 104, 131, 160, 198, 233, 284, 332, 396, 459, 538, 616, 719, 814, 934, 1056, 1203, 1344, 1521, 1692, 1899, 2103, 2343, 2580, 2866, 3139, 3461, 3784, 4156, 4518, 4944, 5360, 5840, 6314, 6852, 7384, 7997
OFFSET
0,7
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} j.
a(n) = A308775(n) - A308733(n) - A308759(n) - A308760(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4*(1 + x + x^2 + x^3 + x^4 + x^5 + 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) | 7 9 15 20 29 ...
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- Wesley Ivan Hurt, Sep 07 2019
MATHEMATICA
Table[Sum[Sum[Sum[j, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
Table[Total[IntegerPartitions[n, {4}][[All, 3]]], {n, 0, 60}] (* Harvey P. Dale, Dec 10 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 22 2019
STATUS
approved