A348919 Sum of the middle parts of the partitions of k into 3 parts for all 0 <= k <= n.
0, 0, 0, 1, 2, 5, 10, 18, 29, 47, 69, 100, 140, 191, 253, 333, 426, 540, 675, 834, 1017, 1234, 1478, 1760, 2080, 2442, 2846, 3305, 3810, 4375, 5000, 5690, 6445, 7281, 8187, 9180, 10260, 11433, 12699, 14077, 15554, 17150, 18865, 20706, 22673, 24788, 27036, 29440, 32000, 34724
Offset: 0
Examples
Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
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n | 3 4 5 6 7 8 9 10 ...
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a(n) | 1 2 5 10 18 29 47 69 ...
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Links
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (1,2,0,-3,-3,3,3,0,-2,-1,1).
Programs
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Mathematica
CoefficientList[Series[-x^3*(x^4 + x^3 + x^2 + x + 1)/((x + 1)^2*(x^2 + x + 1)^2*(x - 1)^5), {x, 0, 49}], x] (* Michael De Vlieger, Nov 05 2021 *)
Formula
a(n) = Sum_{m=1..n} Sum_{k=1..floor(m/3)} Sum_{i=k..floor((m-k)/2)} i.
G.f.: -x^3*(x^4+x^3+x^2+x+1)/((x+1)^2*(x^2+x+1)^2*(x-1)^5). - Alois P. Heinz, Nov 03 2021
a(n) ~ 5*n^4/864. - Stefano Spezia, Nov 04 2021
a(n) = a(n-1)+2*a(n-2)-3*a(n-4)-3*a(n-5)+3*a(n-6)+3*a(n-7)-2*a(n-9)-a(n-10)+a(n-11). - Wesley Ivan Hurt, Nov 19 2021
Comments