A393403 Sum of the smallest parts in the partitions of n into 3 parts whose largest part is odd.

0, 0, 0, 1, 0, 1, 1, 4, 3, 7, 4, 10, 9, 17, 14, 25, 19, 32, 29, 46, 40, 60, 50, 74, 68, 96, 86, 119, 104, 141, 131, 174, 159, 207, 186, 240, 225, 285, 264, 331, 303, 376, 355, 436, 408, 496, 460, 556, 528, 632, 596, 709, 664, 785, 749, 880, 835, 975, 920, 1070, 1025, 1185, 1130, 1301, 1235, 1416, 1361, 1554, 1488, 1692, 1614

Offset

0,8

Links

Table of n, a(n) for n=0..70.

Index entries for sequences related to partitions

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

Formula

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

a(n) = a(n-1) + a(n-4) - a(n-5) + 2*a(n-6) - 2*a(n-7) - 2*a(n-10) + 2*a(n-11) - a(n-12) + a(n-13) + a(n-16) - a(n-17).

a(n) + A393406(n) + A309692(n) = n*A026923(n). - Wesley Ivan Hurt, Feb 17 2026

a(n) ~ n^3/216. - Charles R Greathouse IV, May 31 2026

Example

a(11) = 10; 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 smallest parts of these partitions is 1+1+1+2+2+3 = 10.

Mathematica

Table[Sum[Sum[j*Mod[n - j - k, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
LinearRecurrence[{1, 0, 0, 1, -1, 2, -2, 0, 0, -2, 2, -1, 1, 0, 0, 1, -1}, {0, 0, 0, 1, 0, 1, 1, 4, 3, 7, 4, 10, 9, 17, 14, 25, 19}, 100]

Prog

(PARI) a(n)=(2*n^3+(3+n%2*12)*n^2+12*[0, 3, 0, 3, -4, -1, 0, 3, 0, 3, -4, -1][n%12+1]*n+[0, -53, -28, 135, 16, -133, -108, 55, 80, 27, -92, -25][n%12+1])/432 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A026923, A309692, A393406.

Sequence in context: A340012 A046548 A127752 * A198874 A276760 A365214

Adjacent sequences: A393400 A393401 A393402 * A393404 A393405 A393406

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 13 2026

Status

approved