A309549 - OEIS

%I #20 Aug 20 2024 14:36:13

%S 0,0,0,0,0,0,6,5,10,13,22,27,42,52,76,94,128,156,208,247,314,376,468,

%T 551,674,787,948,1099,1300,1496,1758,2004,2324,2641,3036,3425,3910,

%U 4388,4974,5557,6254,6956,7794,8626,9608,10603,11758,12922,14270,15631

%N Number of odd parts in the partitions of n into 6 parts.

%H Alois P. Heinz, <a href="/A309549/b309549.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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)).

%e 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.

%t 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}]

%t Table[Count[Flatten[IntegerPartitions[n,{6}]],_?(OddQ[#]&)],{n,0,50}] (* _Harvey P. Dale_, Aug 20 2024 *)

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

%K nonn

%O 0,7

%A _Wesley Ivan Hurt_, Aug 07 2019