A308758 Sum of the third largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 1, 1, 2, 4, 7, 9, 15, 20, 29, 38, 51, 64, 86, 104, 131, 160, 198, 233, 284, 332, 396, 459, 538, 616, 719, 814, 934, 1056, 1203, 1344, 1521, 1692, 1899, 2103, 2343, 2580, 2866, 3139, 3461, 3784, 4156, 4518, 4944, 5360, 5840, 6314, 6852, 7384, 7997

Offset

0,7

Links

Table of n, a(n) for n=0..52.

Index entries for sequences related to partitions

Formula

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

a(n) = A308775(n) - A308733(n) - A308759(n) - A308760(n).

Conjectures from Colin Barker, Jun 23 2019: (Start)

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

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.

(End)

Example

Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |      7           9          15          20          29        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 07 2019

Mathematica

Table[Sum[Sum[Sum[j, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
Table[Total[IntegerPartitions[n, {4}][[All, 3]]], {n, 0, 60}] (* Harvey P. Dale, Dec 10 2021 *)

Crossrefs

Cf. A026810, A308733, A308759, A308760, A308775.

Sequence in context: A115162 A278977 A097433 * A110078 A257064 A085800

Adjacent sequences: A308755 A308756 A308757 * A308759 A308760 A308761

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 22 2019

Status

approved