A309715 Number of even parts appearing among the third largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 19, 22, 26, 30, 35, 40, 46, 52, 59, 66, 74, 82, 91, 100, 111, 122, 134, 146, 159, 172, 187, 202, 219, 236, 254, 272, 292, 312, 334, 356, 380, 404, 430, 456, 484, 512, 542, 572, 605, 638, 673, 708, 745

Offset

0,9

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-1) mod 2).

From Colin Barker, Aug 24 2019: (Start)

G.f.: x^7 / ((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) |      2           3           4           5           6        ...
--------------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Sum[Mod[j - 1, 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, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 13}, 60]

Prog

(PARI) concat([0, 0, 0, 0, 0, 0, 0], Vec(x^7 / ((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. A309794, A309796, A309798.

Sequence in context: A116910 A141283 A276828 * A241089 A186445 A080078

Adjacent sequences: A309712 A309713 A309714 * A309716 A309717 A309718

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 13 2019

Status

approved