A309711 Sum of the even parts appearing among the smallest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 8, 10, 14, 16, 24, 28, 36, 44, 54, 62, 76, 86, 106, 120, 142, 162, 188, 210, 242, 268, 310, 342, 388, 430, 482, 528, 590, 642, 718, 780, 862, 938, 1030, 1112, 1218, 1310, 1434, 1540, 1674, 1798, 1946, 2080, 2246, 2394

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 (1,1,0,-1,-1,1,0,2,-2,-2,0,2,2,-2,0,-1,1,1,0,-1,-1,1).

Formula

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

From Colin Barker, Aug 23 2019: (Start)

G.f.: 2*x^8 / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)*(1 + x^4)^2).

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) + 2*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-12) + 2*a(n-13) - 2*a(n-14) - a(n-16) + a(n-17) + a(n-18) - a(n-20) - a(n-21) + a(n-22) for n>21.

(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           2           4           6           8        ...
--------------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Sum[k * Mod[k - 1, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
LinearRecurrence[{1, 1, 0, -1, -1, 1, 0, 2, -2, -2, 0, 2, 2, -2, 0, -1, 1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 8, 10, 14, 16, 24, 28, 36, 44, 54, 62}, 50]

Prog

(PARI) concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(2*x^8 / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)*(1 + x^4)^2) + O(x^60))) \\ Colin Barker, Aug 23 2019

Crossrefs

Cf. A309794, A309796, A309798.

Sequence in context: A118303 A145817 A145809 * A116859 A179999 A286736

Adjacent sequences: A309708 A309709 A309710 * A309712 A309713 A309714

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 13 2019

Status

approved