A309692 - OEIS

A309692

Sum of the odd parts appearing among the largest parts of the partitions of n into 3 parts.

%I #17 Oct 13 2023 17:25:49

%S 0,0,0,1,0,3,3,11,8,20,17,38,33,60,55,95,83,131,124,189,173,248,232,

%T 328,308,416,396,529,496,643,619,795,756,948,909,1134,1089,1332,1287,

%U 1567,1503,1803,1752,2093,2021,2384,2312,2720,2640,3072,2992,3473,3368

%N Sum of the odd parts appearing among the largest parts of the partitions of n into 3 parts.

%H Colin Barker, <a href="/A309692/b309692.txt">Table of n, a(n) for n = 0..1000</a>

%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_{i=j..floor((n-j)/2)} (n-i-j) * ((n-i-j) mod 2).

%F From _Colin Barker_, Aug 23 2019: (Start)

%F G.f.: x^3*(1 - x + 4*x^2 - x^3 + 10*x^4 - 2*x^5 + 14*x^6 - 3*x^7 + 14*x^8 - 3*x^9 + 8*x^10 + 3*x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^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) for n>18.

%F (End)

%e Figure 1: The partitions of n into 3 parts for n = 3, 4, ...

%e 1+1+8

%e 1+1+7 1+2+7

%e 1+2+6 1+3+6

%e 1+1+6 1+3+5 1+4+5

%e 1+1+5 1+2+5 1+4+4 2+2+6

%e 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5

%e 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4

%e 1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...

%e -----------------------------------------------------------------------

%e n | 3 4 5 6 7 8 9 10 ...

%e -----------------------------------------------------------------------

%e a(n) | 1 0 3 3 11 8 20 17 ...

%e -----------------------------------------------------------------------

%t Table[Sum[Sum[ (n - i - j) * Mod[n - i - j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]

%t 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, 3, 3, 11, 8, 20, 17, 38, 33, 60, 55, 95, 83, 131, 124}, 80]

%t Table[Total[Select[IntegerPartitions[n,{3}][[;;,1]],OddQ]],{n,0,60}] (* _Harvey P. Dale_, Oct 13 2023 *)

%o (PARI) concat([0,0,0], Vec(x^3*(1 - x + 4*x^2 - x^3 + 10*x^4 - 2*x^5 + 14*x^6 - 3*x^7 + 14*x^8 - 3*x^9 + 8*x^10 + 3*x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^40))) \\ _Colin Barker_, Aug 23 2019

%Y Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309694.

%K nonn,easy

%O 0,6

%A _Wesley Ivan Hurt_, Aug 12 2019