A309831 Number of even parts appearing among the smallest parts of the partitions of n into 5 parts.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 24, 28, 36, 42, 52, 60, 73, 83, 99, 112, 132, 148, 172, 192, 221, 245, 279, 308, 348, 382, 429, 469, 523, 570, 632, 686, 757, 819, 899, 970, 1061, 1141, 1243, 1334, 1448, 1550, 1677, 1791, 1932
Offset: 0
Examples
Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
1+1+1+1+10
1+1+1+2+9
1+1+1+3+8
1+1+1+4+7
1+1+1+5+6
1+1+1+1+9 1+1+2+2+8
1+1+1+2+8 1+1+2+3+7
1+1+1+3+7 1+1+2+4+6
1+1+1+4+6 1+1+2+5+5
1+1+1+5+5 1+1+3+3+6
1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
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n | 10 11 12 13 14 ...
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a(n) | 1 1 2 3 5 ...
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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 (1,1,0,0,-2,0,0,1,1,0,-1,-1,0,0,2,0,0,-1,-1,1).
Programs
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Mathematica
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, 0, -1, -1, 0, 0, 2, 0, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18}, 50] -
PARI
concat([0,0,0,0,0,0,0,0,0,0], Vec(x^10 / ((1 - x)^5*(1 + x)^3*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^80))) \\ Colin Barker, Oct 10 2019
Formula
a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} ((l-1) mod 2).
From Colin Barker, Aug 19 2019: (Start)
G.f.: x^10 / ((1 - x)^5*(1 + x)^3*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-11) - a(n-12) + 2*a(n-15) - a(n-18) - a(n-19) + a(n-20) for n>19.
(End) [Recurrence verified by Wesley Ivan Hurt, Aug 24 2019]