A309793 Number of odd parts appearing among the second largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 13, 17, 20, 24, 27, 32, 36, 42, 47, 54, 60, 68, 75, 85, 93, 103, 112, 124, 135, 149, 161, 176, 189, 205, 220, 239, 256, 276, 294, 316, 336, 360, 382, 408, 432, 460, 486, 517, 545, 577, 607, 642, 675, 713, 748

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)} (i mod 2).

From Colin Barker, Aug 18 2019: (Start)

G.f.: x^4*(1 - x + x^4) / ((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) [Recurrence verified by Wesley Ivan Hurt, Aug 24 2019]

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           5           6           8        ...
--------------------------------------------------------------------------

Mathematica

LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 13}, 50]

Prog

(PARI) concat([0, 0, 0, 0], Vec(x^4*(1 - x + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)) + O(x^50))) \\ Colin Barker, Oct 10 2019

Crossrefs

Cf. A309795, A309797, A026928.

Sequence in context: A191877 A277124 A094820 * A093689 A097702 A347912

Adjacent sequences: A309790 A309791 A309792 * A309794 A309795 A309796

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 17 2019

Status

approved