A308824 Sum of the fourth largest parts in the partitions of n into 5 parts.

0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 13, 18, 27, 36, 50, 64, 86, 109, 140, 175, 220, 269, 331, 399, 486, 577, 689, 811, 959, 1119, 1305, 1508, 1747, 2003, 2300, 2617, 2984, 3376, 3821, 4300, 4839, 5415, 6060, 6749, 7521, 8337, 9243, 10207, 11273, 12404, 13641

Offset: 0

Author

Wesley Ivan Hurt, Jun 26 2019

Keywords

• nonn

Examples

			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) |      9          13          18          27          36        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 08 2019
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A026811, A308822, A308823, A308825, A308826, A308827.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[k, {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}]

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)} k.

a(n) = A308822(n) - A308823(n) - A308825(n) - A308826(n) - A308827(n).

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

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

a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-4) - 4*a(n-6) + a(n-8) - 3*a(n-9) + 4*a(n-10) + 4*a(n-11) - 3*a(n-12) + a(n-13) - 4*a(n-15) + 2*a(n-17) - a(n-18) + a(n-19) + a(n-20) - a(n-21) for n>20.

(End)