A309712 Number of odd parts appearing among the third largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 6, 9, 10, 13, 14, 18, 20, 25, 28, 34, 37, 44, 48, 56, 61, 70, 76, 87, 94, 106, 114, 127, 136, 151, 162, 179, 191, 209, 222, 242, 257, 279, 296, 320, 338, 364, 384, 412, 434, 464, 488, 521, 547, 582, 610, 647, 677, 717, 750

Offset

0,7

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

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

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (j mod 2).

From Colin Barker, Aug 24 2019: (Start)

G.f.: x^4*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)).

a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) - a(n-14) + 2*a(n-15) - 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) |      3           3           5           6           9        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 04 2019

Mathematica

Table[Sum[Sum[Sum[Mod[j, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 6, 9, 10, 13, 14}, 60] (* Wesley Ivan Hurt, Sep 04 2019 *)

Prog

(PARI) concat([0, 0, 0, 0], Vec(x^4*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)) + O(x^70))) \\ Colin Barker, Aug 24 2019

Crossrefs

Cf. A309711, A309715.

Sequence in context: A240201 A274158 A020999 * A079955 A192928 A136417

Adjacent sequences: A309709 A309710 A309711 * A309713 A309714 A309715

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 13 2019

Status

approved