A308733 Sum of the smallest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 1, 1, 2, 3, 6, 7, 11, 14, 21, 25, 34, 41, 55, 64, 81, 95, 119, 136, 165, 189, 227, 256, 301, 339, 396, 441, 507, 564, 645, 711, 804, 885, 996, 1089, 1215, 1326, 1474, 1600, 1766, 1914, 2106, 2272, 2486, 2678, 2922, 3136, 3406, 3650, 3955, 4225, 4560

Offset

0,7

Links

Table of n, a(n) for n=0..54.

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)} k.

a(n) = A308775(n) - A308758(n) - A308759(n) - A308760(n).

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

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

a(n) = a(n-1) + a(n-2) + a(n-4) - 3*a(n-5) - a(n-6) + a(n-8) + 3*a(n-9) - a(n-10) - a(n-12) - a(n-13) + a(n-14) for n>13.

(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) |      6           7          11          14          21        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 07 2019

Mathematica

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

Crossrefs

Cf. A026810, A308758, A308759, A308760, A308775.

Sequence in context: A008765 A018468 A117115 * A363046 A049196 A284743

Adjacent sequences: A308730 A308731 A308732 * A308734 A308735 A308736

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 22 2019

Status

approved