A309708 Sum of the odd parts appearing among the smallest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 13, 15, 20, 25, 31, 36, 45, 51, 65, 74, 89, 103, 121, 136, 159, 177, 208, 231, 265, 296, 335, 369, 416, 455, 514, 561, 625, 684, 756, 820, 904, 976, 1076, 1160, 1268, 1368, 1488, 1596, 1732, 1852, 2009, 2145, 2314, 2471

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 (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 mod 2).

From Colin Barker, Aug 23 2019: (Start)

G.f.: x^4*(1 + 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) |      4           5           7           8          13        ...
--------------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Sum[k * Mod[k, 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, 1, 1, 2, 3, 4, 5, 7, 8, 13, 15, 20, 25, 31, 36, 45, 51, 65, 74}, 80] (* Wesley Ivan Hurt, Sep 04 2019 *)

Prog

(PARI) concat([0, 0, 0, 0], Vec(x^4*(1 + 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. A309711, A309715.

Sequence in context: A091997 A124168 A379972 * A285929 A309880 A054762

Adjacent sequences: A309705 A309706 A309707 * A309709 A309710 A309711

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 13 2019

Status

approved