A309549 Number of odd parts in the partitions of n into 6 parts.
0, 0, 0, 0, 0, 0, 6, 5, 10, 13, 22, 27, 42, 52, 76, 94, 128, 156, 208, 247, 314, 376, 468, 551, 674, 787, 948, 1099, 1300, 1496, 1758, 2004, 2324, 2641, 3036, 3425, 3910, 4388, 4974, 5557, 6254, 6956, 7794, 8626, 9608, 10603, 11758, 12922, 14270, 15631
Offset: 0
Keywords
Examples
a(9) = 13 because the 3 possible partitions into 6 parts, 1+1+1+1+1+4, 1+1+1+1+2+3, and 1+1+1+2+2+2 contain in total 13 odd numbers.
Links
Programs
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Mathematica
Table[Sum[Sum[Sum[Sum[Sum[Mod[i, 2] + Mod[j, 2] + Mod[k, 2] + Mod[l, 2] + Mod[m, 2] + Mod[n - i - j - k - l - m, 2], {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}] Table[Count[Flatten[IntegerPartitions[n,{6}]],?(OddQ[#]&)],{n,0,50}] (* _Harvey P. Dale, Aug 20 2024 *)
Formula
a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} ((i mod 2) + (j mod 2) + (k mod 2) + (l mod 2) + (m mod 2) + ((n-i-j-k-l-m) mod 2)).