A308760 Sum of the largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 1, 2, 5, 9, 17, 25, 41, 57, 84, 112, 154, 197, 262, 325, 414, 506, 629, 751, 915, 1078, 1289, 1501, 1767, 2034, 2370, 2701, 3108, 3519, 4014, 4506, 5100, 5691, 6393, 7095, 7917, 8739, 9703, 10658, 11765, 12876, 14150, 15418, 16874, 18324, 19974

Offset

0,6

Links

Table of n, a(n) for n=0..48.

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)} (n-i-j-k).

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

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

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).

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) |     17          25          41          57          84        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 07 2019

Mathematica

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}]

Crossrefs

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

Sequence in context: A167887 A023603 A334087 * A342854 A062492 A268346

Adjacent sequences: A308757 A308758 A308759 * A308761 A308762 A308763

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 22 2019

Status

approved