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
Examples
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 ...
--------------------------------------------------------------------------
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).
Programs
-
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] -
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
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]