A309879 Number of odd parts appearing among the fourth largest parts of the partitions of n into 5 parts.

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 11, 14, 18, 22, 28, 33, 40, 47, 56, 65, 77, 89, 104, 119, 137, 155, 177, 199, 225, 252, 283, 315, 352, 389, 432, 476, 525, 576, 633, 691, 756, 823, 897, 973, 1057, 1143, 1237, 1334, 1439, 1547, 1665, 1786, 1917, 2052

Offset

0,8

Links

Table of n, a(n) for n=0..56.

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 mod 2).

Conjectures from Colin Barker, Aug 22 2019: (Start)

G.f.: x^5*(1 - x + x^2)*(1 - x^3 + x^6) / ((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) [Conjectures verified by Wesley Ivan Hurt, Aug 24 2019]

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) |      5           7           8          11          14        ...
--------------------------------------------------------------------------

Mathematica

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, 1, 1, 2, 3, 4, 5, 7, 8,
  11, 14, 18, 22, 28, 33, 40, 47, 56, 65}, 50]

Prog

(PARI) Vec(x^5*(1-x+x^2)*(1-x^3+x^6)/((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)) + O(x^70)) \\ Jinyuan Wang, Feb 28 2020

Crossrefs

Cf. A309880, A309881, A309882.

Sequence in context: A116470 A115649 A191168 * A191166 A366064 A238484

Adjacent sequences: A309876 A309877 A309878 * A309880 A309881 A309882

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 21 2019

Status

approved