A309516 -id:A309516 - cp's OEIS frontend

A309543 Number of odd parts in the partitions of n into 5 parts.

0, 0, 0, 0, 0, 5, 4, 8, 10, 17, 20, 32, 38, 56, 66, 90, 104, 137, 158, 200, 230, 285, 324, 393, 444, 530, 594, 697, 778, 905, 1004, 1153, 1274, 1450, 1594, 1802, 1972, 2213, 2414, 2690, 2924, 3242, 3512, 3873, 4184, 4595, 4948, 5410, 5812, 6330, 6784, 7362

Offset: 0

Wesley Ivan Hurt, Aug 06 2019

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     20          32          38          56          66        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 12 2019

Index entries for sequences related to partitions

Cf. A309516.

Table[Sum[Sum[Sum[Sum[Mod[i, 2] + Mod[j, 2] + Mod[k, 2] + Mod[l, 2] + Mod[n - i - j - k - l, 2], {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-1)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} ((i mod 2) + (j mod 2) + (k mod 2) + (l mod 2) + ((n-i-j-k-l) mod 2)).

G.f.: -x^5*(2*x^11-x^10+2*x^8+4*x^6-4*x^5+5*x^4-2*x^3+5*x^2-6*x+5) / ((x^2+1) *(x^2+x+1) *(x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) *(x^4+1) *(x+1)^3 *(x-1)^5). - Alois P. Heinz, Aug 07 2019

A309549 Number of odd parts in the partitions of n into 6 parts.

Original entry on oeis.org

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

Wesley Ivan Hurt, Aug 07 2019

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

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

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 *)

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)).

cp's OEIS Frontend

A309543 Number of odd parts in the partitions of n into 5 parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula