A393405 - OEIS

%I #12 Jul 25 2026 12:11:24

%S 0,0,0,1,1,3,3,6,6,13,13,22,22,33,33,51,51,72,72,96,96,130,130,168,

%T 168,210,210,265,265,325,325,390,390,471,471,558,558,651,651,763,763,

%U 882,882,1008,1008,1156,1156,1312,1312,1476,1476,1665,1665,1863,1863,2070,2070,2305,2305,2550,2550,2805,2805,3091,3091,3388,3388

%N Sum of the middle parts in the partitions of n into 3 parts whose smallest part is odd.

%H Harvey P. Dale, <a href="/A393405/b393405.txt">Table of n, a(n) for n = 0..1000</a>

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

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,2,-2,-2,2,0,0,-1,1,1,-1).

%F a(n) = Sum_{j=1..floor(n/3)} Sum_{k=j..floor((n-j)/2)} k * (j mod 2).

%F a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-6) - 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - a(n-12) + a(n-13) + a(n-14) - a(n-15).

%F a(n) + A309684(n) + A393410(n) = n*A309683(n). - _Wesley Ivan Hurt_, Feb 17 2026

%F a(n) ~ 5*n^3/432. - _Charles R Greathouse IV_, May 31 2026

%e a(10) = 13; There are 5 partitions of 10 into 3 parts whose smallest part is odd: (8,1,1), (7,2,1), (6,3,1), (5,4,1) and (4,3,3). The sum of the middle parts of these partitions is 1+2+3+4+3 = 13.

%t Table[Sum[Sum[k*Mod[j, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]

%t (* Alternative: *)

%t LinearRecurrence[{1, 1, -1, 0, 0, 2, -2, -2, 2, 0, 0, -1, 1, 1, -1}, {0, 0, 0, 1, 1, 3, 3, 6, 6, 13, 13, 22, 22, 33, 33}, 100]

%t (* Alternative: *)

%t Table[Total[Select[IntegerPartitions[n,{3}],OddQ[#[[3]]]&][[;;,2]]],{n,0,70}] (* _Harvey P. Dale_, Jul 25 2026 *)

%o (PARI) a(n)=(5*n^3+(6+n%2*15)*n^2+3*[0,-7,-16,9,0,9][n%6+1]*n+[0,-5,32,27,16,11][n%6+1])/432 \\ _Charles R Greathouse IV_, May 31 2026

%Y Cf. A309683, A309684, A393410.

%K nonn,easy,changed

%O 0,6

%A _Wesley Ivan Hurt_, Feb 13 2026