A309882 Sum of the even parts appearing among the fourth largest parts of the partitions of n into 5 parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 10, 14, 18, 24, 30, 40, 52, 68, 88, 110, 136, 166, 198, 240, 286, 340, 404, 478, 560, 652, 754, 872, 1000, 1146, 1308, 1488, 1686, 1908, 2148, 2416, 2708, 3028, 3376, 3758, 4168, 4616, 5098, 5630, 6200, 6816, 7482, 8198

Offset

0,10

Links

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

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -1).

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 * ((k-1) mod 2).

a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 2*a(n-8) - 2*a(n-9) + 6*a(n-10) - 10*a(n-11) + 12*a(n-12) - 12*a(n-13) + 12*a(n-14) - 12*a(n-15) + 11*a(n-16) - 9*a(n-17) + 4*a(n-18) + 4*a(n-19) - 9*a(n-20) + 11*a(n-21) - 12*a(n-22) + 12*a(n-23) - 12*a(n-24) + 12*a(n-25) - 10*a(n-26) + 6*a(n-27) - 2*a(n-28) - 2*a(n-29) + 4*a(n-30) - 4*a(n-31) + 4*a(n-32) - 4*a(n-33) + 4*a(n-34) - 4*a(n-35) + 3*a(n-36) - a(n-37) for n > 36.

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) |      4           6          10          14          18        ...
--------------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Sum[Sum[k * Mod[k - 1, 2], {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, 50}]
LinearRecurrence[{3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12,
  12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2,
  4, -4, 4, -4, 4, -4, 3, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6,
  10, 14, 18, 24, 30, 40, 52, 68, 88, 110, 136, 166, 198, 240, 286,
  340, 404, 478, 560, 652, 754, 872, 1000, 1146, 1308}, 50]

Crossrefs

Cf. A309879, A309880, A309881.

Sequence in context: A059254 A024518 A128422 * A098380 A007782 A035501

Adjacent sequences: A309879 A309880 A309881 * A309883 A309884 A309885

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 21 2019

Status

approved