A393409 - OEIS

%I #7 May 31 2026 21:56:55

%S 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,

%T 174,210,216,265,276,325,326,390,407,471,477,558,570,651,663,763,782,

%U 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

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,1,-1,3,-3,2,-2,-2,2,-3,3,-1,1,1,-1,1,-1).

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

%F 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).

%F a(n) + A393404(n) + A309694(n) = n*A026927(n). - _Wesley Ivan Hurt_, Feb 17 2026

%F a(n) ~ 5*n^3/432. - _Charles R Greathouse IV_, May 31 2026

%e 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.

%t Table[Sum[Sum[k*Mod[n - k - j + 1, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]

%t 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]

%o (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

%Y Cf. A026927, A309694, A393404.

%K nonn,easy

%O 0,6

%A _Wesley Ivan Hurt_, Feb 13 2026