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A309715
Number of even parts appearing among the third largest parts of the partitions of n into 4 parts.
6
0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 19, 22, 26, 30, 35, 40, 46, 52, 59, 66, 74, 82, 91, 100, 111, 122, 134, 146, 159, 172, 187, 202, 219, 236, 254, 272, 292, 312, 334, 356, 380, 404, 430, 456, 484, 512, 542, 572, 605, 638, 673, 708, 745
OFFSET
0,9
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-1) mod 2).
From Colin Barker, Aug 24 2019: (Start)
G.f.: x^7 / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)).
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) - a(n-14) + 2*a(n-15) - 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) | 2 3 4 5 6 ...
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MATHEMATICA
Table[Sum[Sum[Sum[Mod[j - 1, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 13}, 60]
PROG
(PARI) concat([0, 0, 0, 0, 0, 0, 0], Vec(x^7 / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)) + O(x^70))) \\ Colin Barker, Aug 24 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 13 2019
STATUS
approved