A308823 Sum of the smallest parts of the partitions of n into 5 parts.

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 15, 21, 28, 38, 48, 62, 78, 98, 122, 149, 181, 219, 262, 314, 370, 436, 510, 595, 691, 797, 916, 1050, 1198, 1365, 1545, 1747, 1968, 2212, 2480, 2771, 3089, 3437, 3814, 4227, 4669, 5151, 5670, 6232, 6838, 7487, 8185, 8936

Offset

0,8

Links

Table of n, a(n) for n=0..53.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} l.

a(n) = A308822(n) - A308824(n) - A308825(n) - A308826(n) - A308827(n).

Conjectures from Colin Barker, Jun 30 2019: (Start)

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

a(n) = a(n-1) + a(n-2) - 2*a(n-6) - 2*a(n-7) + a(n-8) + a(n-9) + 2*a(n-10) + a(n-11) + a(n-12) - 2*a(n-13) - 2*a(n-14) + a(n-18) + a(n-19) - a(n-20) for n>19.

(End)

Example

Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      8          11          15          21          28        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 08 2019

Mathematica

Table[Sum[Sum[Sum[Sum[l, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]
Table[Total[IntegerPartitions[n, {5}][[;; , 5]]], {n, 0, 60}] (* Harvey P. Dale, Nov 20 2024 *)

Crossrefs

Cf. A026811, A308822, A308824, A308825, A308826, A308827.

Sequence in context: A281706 A071424 A008762 * A363047 A101018 A320593

Adjacent sequences: A308820 A308821 A308822 * A308824 A308825 A308826

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 26 2019

Status

approved