A393405 Sum of the middle parts in the partitions of n into 3 parts whose smallest part is odd.

0, 0, 0, 1, 1, 3, 3, 6, 6, 13, 13, 22, 22, 33, 33, 51, 51, 72, 72, 96, 96, 130, 130, 168, 168, 210, 210, 265, 265, 325, 325, 390, 390, 471, 471, 558, 558, 651, 651, 763, 763, 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

Offset

0,6

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,2,-2,-2,2,0,0,-1,1,1,-1).

Formula

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

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).

a(n) + A309684(n) + A393410(n) = n*A309683(n). - Wesley Ivan Hurt, Feb 17 2026

a(n) ~ 5*n^3/432. - Charles R Greathouse IV, May 31 2026

Example

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.

Mathematica

Table[Sum[Sum[k*Mod[j, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
(* Alternative: *)
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]
(* Alternative: *)
Table[Total[Select[IntegerPartitions[n, {3}], OddQ[#[[3]]]&][[;; , 2]]], {n, 0, 70}] (* Harvey P. Dale, Jul 25 2026 *)

Prog

(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

Crossrefs

Cf. A309683, A309684, A393410.

Sequence in context: A240204 A200905 A280510 * A292128 A117775 A379358

Adjacent sequences: A393402 A393403 A393404 * A393406 A393407 A393408

Keyword

nonn,easy,changed

Author

Wesley Ivan Hurt, Feb 13 2026

Status

approved