A393409 Sum of the middle parts in the partitions of n into 3 parts whose largest part is even.

0, 0, 0, 0, 1, 2, 3, 2, 6, 9, 13, 13, 22, 24, 33, 35, 51, 56, 72, 71, 96, 105, 130, 132, 168, 174, 210, 216, 265, 276, 325, 326, 390, 407, 471, 477, 558, 570, 651, 663, 763, 782, 882, 887, 1008, 1035, 1156, 1168, 1312, 1332, 1476, 1496, 1665, 1694, 1863, 1874, 2070, 2109, 2305, 2325, 2550, 2580, 2805, 2835, 3091, 3132, 3388

Offset

0,6

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-k-j+1) 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) + A393404(n) + A309694(n) = n*A026927(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 largest part is even: (8,1,1), (6,3,1), (6,2,2), (4,4,2) and (4,3,3). The sum of the middle parts of these partitions is 1+3+2+4+3 = 13.

Mathematica

Table[Sum[Sum[k*Mod[n - k - j + 1, 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, 0, 1, 2, 3, 2, 6, 9, 13, 13, 22, 24, 33, 35, 51, 56, 72}, 100]

Prog

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

Crossrefs

Cf. A026927, A309694, A393404.

Sequence in context: A153984 A382824 A025502 * A110777 A087454 A059446

Adjacent sequences: A393406 A393407 A393408 * A393410 A393411 A393412

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 13 2026

Status

approved