login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Sum of the largest parts of the partitions of n into 4 parts.
4

%I #21 Sep 07 2019 09:47:16

%S 0,0,0,0,1,2,5,9,17,25,41,57,84,112,154,197,262,325,414,506,629,751,

%T 915,1078,1289,1501,1767,2034,2370,2701,3108,3519,4014,4506,5100,5691,

%U 6393,7095,7917,8739,9703,10658,11765,12876,14150,15418,16874,18324,19974

%N Sum of the largest parts of the partitions of n into 4 parts.

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

%F a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (n-i-j-k).

%F a(n) = A308775(n) - A308733(n) - A308758(n) - A308759(n).

%F Conjectures from _Colin Barker_, Jun 23 2019: (Start)

%F G.f.: x^4*(1 + 2*x + 4*x^2 + 5*x^3 + 6*x^4 + 4*x^5 + 3*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).

%F a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.

%F (End)

%e Figure 1: The partitions of n into 4 parts for n = 8, 9, ..

%e 1+1+1+9

%e 1+1+2+8

%e 1+1+3+7

%e 1+1+4+6

%e 1+1+1+8 1+1+5+5

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

%e 1+1+1+7 1+1+3+6 1+2+3+6

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

%e 1+1+3+5 1+2+2+6 1+3+3+5

%e 1+1+1+6 1+1+4+4 1+2+3+5 1+3+4+4

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

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

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

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

%e 2+2+2+2 2+2+2+3 2+2+3+3 2+3+3+3 3+3+3+3

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

%e n | 8 9 10 11 12 ...

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

%e a(n) | 17 25 41 57 84 ...

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

%e - _Wesley Ivan Hurt_, Sep 07 2019

%t Table[Sum[Sum[Sum[n - i - j - k, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

%Y Cf. A001318, A026810, A308265, A308733, A308758, A308759, A308775.

%K nonn

%O 0,6

%A _Wesley Ivan Hurt_, Jun 22 2019