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A309881 Number of even parts appearing among the fourth largest parts of the partitions of n into 5 parts. 3

%I #18 Sep 13 2019 22:37:08

%S 0,0,0,0,0,0,0,0,0,1,2,3,5,7,9,12,15,19,24,30,37,45,54,64,75,88,102,

%T 118,136,156,178,202,228,257,288,322,359,399,442,489,539,593,651,713,

%U 779,850,925,1005,1090,1181,1277,1379,1487,1601,1721,1848,1981,2122

%N Number of even parts appearing among the fourth largest parts of the partitions of n into 5 parts.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1,0,0,1,-2,2,-3,3,-2,2,-1,0,0,-1,2,-1,1,-2,1).

%F 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)} ((k-1) mod 2).

%F From _Colin Barker_, Aug 22 2019: (Start)

%F G.f.: x^9 / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x^4)*(1 + x + x^2 + x^3 + x^4)).

%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) + a(n-8) - 2*a(n-9) + 2*a(n-10) - 3*a(n-11) + 3*a(n-12) - 2*a(n-13) + 2*a(n-14) - a(n-15) - a(n-18) + 2*a(n-19) - a(n-20) + a(n-21) - 2*a(n-22) + a(n-23) for n>22.

%F (End)

%e Figure 1: The partitions of n into 5 parts for n = 10, 11, ..

%e 1+1+1+1+10

%e 1+1+1+2+9

%e 1+1+1+3+8

%e 1+1+1+4+7

%e 1+1+1+5+6

%e 1+1+1+1+9 1+1+2+2+8

%e 1+1+1+2+8 1+1+2+3+7

%e 1+1+1+3+7 1+1+2+4+6

%e 1+1+1+4+6 1+1+2+5+5

%e 1+1+1+5+5 1+1+3+3+6

%e 1+1+1+1+8 1+1+2+2+7 1+1+3+4+5

%e 1+1+1+2+7 1+1+2+3+6 1+1+4+4+4

%e 1+1+1+3+6 1+1+2+4+5 1+2+2+2+7

%e 1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6

%e 1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5

%e 1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e --------------------------------------------------------------------------

%e n | 10 11 12 13 14 ...

%e --------------------------------------------------------------------------

%e a(n) | 2 3 5 7 9 ...

%e --------------------------------------------------------------------------

%t Table[Sum[Sum[Sum[Sum[Mod[k - 1, 2], {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

%t LinearRecurrence[{2, -1, 1, -2, 1, 0, 0, 1, -2, 2, -3, 3, -2, 2, -1,

%t 0, 0, -1, 2, -1, 1, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 5,

%t 7, 9, 12, 15, 19, 24, 30, 37, 45, 54}, 50]

%Y Cf. A309879, A309880, A309882.

%K nonn,easy

%O 0,11

%A _Wesley Ivan Hurt_, Aug 21 2019

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