A309881 Number of 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, 1, 2, 3, 5, 7, 9, 12, 15, 19, 24, 30, 37, 45, 54, 64, 75, 88, 102, 118, 136, 156, 178, 202, 228, 257, 288, 322, 359, 399, 442, 489, 539, 593, 651, 713, 779, 850, 925, 1005, 1090, 1181, 1277, 1379, 1487, 1601, 1721, 1848, 1981, 2122

Offset

0,11

Links

Table of n, a(n) for n=0..57.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1,0,0,1,-2,2,-3,3,-2,2,-1,0,0,-1,2,-1,1,-2,1).

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-1) mod 2).

From Colin Barker, Aug 22 2019: (Start)

G.f.: x^9 / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x^4)*(1 + x + x^2 + x^3 + x^4)).

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) + a(n-8) - 2*a(n-9) + 2*a(n-10) - 3*a(n-11) + 3*a(n-12) - 2*a(n-13) + 2*a(n-14) - a(n-15) - a(n-18) + 2*a(n-19) - a(n-20) + a(n-21) - 2*a(n-22) + a(n-23) for n>22.

(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) |      2           3           5           7           9        ...
--------------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Sum[Sum[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[{2, -1, 1, -2, 1, 0, 0, 1, -2, 2, -3, 3, -2, 2, -1,
  0, 0, -1, 2, -1, 1, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 5,
  7, 9, 12, 15, 19, 24, 30, 37, 45, 54}, 50]

Crossrefs

Cf. A309879, A309880, A309882.

Sequence in context: A137285 A062441 A059290 * A293230 A133231 A235111

Adjacent sequences: A309878 A309879 A309880 * A309882 A309883 A309884

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 21 2019

Status

approved