A308265 -id:A308265 - cp's OEIS frontend

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

Wesley Ivan Hurt, Jun 22 2019

nonn

			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

Index entries for sequences related to partitions

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

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}]

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)

A308266 Sum of the middle parts in the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 8, 11, 18, 22, 31, 40, 51, 62, 80, 93, 114, 135, 159, 183, 217, 244, 282, 320, 362, 404, 459, 505, 565, 625, 690, 755, 836, 906, 993, 1080, 1173, 1266, 1378, 1477, 1596, 1715, 1841, 1967, 2115, 2248, 2404, 2560, 2724, 2888, 3077, 3249, 3447

Offset: 1

Wesley Ivan Hurt, May 17 2019

nonn

			Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
                                                          1+1+8
                                                   1+1+7  1+2+7
                                                   1+2+6  1+3+6
                                            1+1+6  1+3+5  1+4+5
                                     1+1+5  1+2+5  1+4+4  2+2+6
                              1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
                       1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
         1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     1      1      3      5      8     11     18     22      ...
-----------------------------------------------------------------------

Index entries for sequences related to partitions

Cf. A308265.

Table[Sum[Sum[i, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
Table[Total[IntegerPartitions[n,{3}][[All,2]]],{n,60}] (* Harvey P. Dale, Jul 09 2020 *)

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} i.
Conjectures from Colin Barker, Jul 16 2019: (Start)
G.f.: x^3*(1 + x + x^2 + x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 4*a(n-5) - a(n-6) + 2*a(n-7) + 2*a(n-8) - a(n-10) for n>10.
(End)

cp's OEIS Frontend

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

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