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

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Index entries for sequences related to partitions

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)).

Example

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.

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 *)

Crossrefs

Cf. A309511, A309516, A309543, A309622, A309628, A309656, A309660.

Sequence in context: A229983 A251859 A100884 * A187892 A046615 A103132

Adjacent sequences: A309546 A309547 A309548 * A309550 A309551 A309552

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 07 2019

Status

approved