A393406 Sum of the middle parts in the partitions of n into 3 parts whose largest part is odd.

0, 0, 0, 1, 0, 1, 2, 6, 5, 9, 9, 18, 18, 27, 29, 45, 42, 58, 63, 88, 87, 112, 114, 150, 152, 188, 194, 243, 240, 289, 300, 364, 365, 429, 435, 516, 522, 603, 615, 715, 714, 814, 833, 954, 959, 1080, 1092, 1236, 1248, 1392, 1412, 1581, 1584, 1753, 1782, 1978, 1989, 2185, 2205, 2430, 2450, 2675, 2705, 2961, 2970, 3226, 3267

Offset

0,7

Links

Table of n, a(n) for n=0..66.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,1,-1,3,-3,2,-2,-2,2,-3,3,-1,1,1,-1,1,-1).

Formula

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

a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + 3*a(n-6) - 3*a(n-7) + 2*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14) + a(n-15) + a(n-16) - a(n-17) + a(n-18) - a(n-19).

a(n) + A393403(n) + A309692(n) = n*A026923(n). - Wesley Ivan Hurt, Feb 17 2026

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

Example

a(11) = 18; There are 6 partitions of 11 into 3 parts whose largest part is odd: (9,1,1), (7,3,1), (5,5,1), (7,2,2), (5,4,2) and (5,3,3). The sum of the middle parts of these partitions is 1+3+5+2+4+3 = 18.

Mathematica

Table[Sum[Sum[k*Mod[n - j - k, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
LinearRecurrence[{1, -1, 1, 1, -1, 3, -3, 2, -2, -2, 2, -3, 3, -1, 1, 1, -1, 1, -1}, {0, 0, 0, 1, 0, 1, 2, 6, 5, 9, 9, 18, 18, 27, 29, 45, 42, 58, 63}, 100]

Prog

(PARI) a(n)=(5*n^3+(-6+n%2*12)*n^2+[0, -27, 0, 81, -48, -75, 0, 81, 0, -27, -48, 33][n%12+1]*n+16*[0, 1, -1, 0, -2, 2, 0, 1, -1, 0, -2, 2][n%12+1])/432 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A026923, A309692, A393403.

Sequence in context: A021380 A195488 A092744 * A077174 A211201 A359035

Adjacent sequences: A393403 A393404 A393405 * A393407 A393408 A393409

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 13 2026

Status

approved