A392475 Sum of the largest parts in the partitions of n into 3 parts whose middle part is even.

0, 0, 0, 0, 0, 2, 5, 7, 9, 15, 22, 30, 39, 51, 64, 78, 93, 117, 143, 165, 188, 222, 258, 296, 336, 382, 430, 480, 532, 600, 671, 735, 801, 885, 972, 1062, 1155, 1257, 1362, 1470, 1581, 1715, 1853, 1981, 2112, 2268, 2428, 2592, 2760, 2940, 3124, 3312, 3504, 3726, 3953, 4167, 4385, 4635, 4890, 5150, 5415, 5695, 5980, 6270

Offset

0,6

Links

Table of n, a(n) for n=0..63.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,1,-4,6,-8,6,-4,1,2,-3,4,-3,2,-1).

Formula

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

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

a(n) + A392567(n) + A309690(n) = n*A309689(n). - Wesley Ivan Hurt, Feb 17 2026

a(n) ~ 11*n^3/432. - Charles R Greathouse IV, May 31 2026

Example

a(10) = 22; There are 4 partitions of 10 into 3 parts whose middle part is even: (7,2,1), (5,4,1), (6,2,2) and (4,4,2). The sum of the largest parts of these partitions is 7+5+6+4 = 22.

Mathematica

Table[Sum[Sum[(n - j - k)*Mod[k + 1, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
LinearRecurrence[{2, -3, 4, -3, 2, 1, -4, 6, -8, 6, -4, 1, 2, -3, 4, -3, 2, -1}, {0, 0, 0, 0, 0, 2, 5, 7, 9, 15, 22, 30, 39, 51, 64, 78, 93, 117}, 100]

Prog

(PARI) a(n)=(11*n^3-9*n^2+[-72, -51, -60, -99, -168, -51, 36, -51, -168, -99, -60, -51][n%12+1]*n+[0, 49, 68, 81, 112, -31, -108, 49, 176, 81, 4, -31][n%12+1])/432 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A309689, A309690, A392567.

Sequence in context: A287363 A253275 A093417 * A286162 A286164 A211167

Adjacent sequences: A392472 A392473 A392474 * A392476 A392477 A392478

Keyword

nonn,easy,changed

Author

Wesley Ivan Hurt, Feb 13 2026

Status

approved