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For k = 6, the diagram 1 represents the partitions of 6. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y], see below:
.
. j Diagram 1 Partitions Diagram 2
. _ _ _ _ _ _ _ _ _ _ _ _
. 11 |_ _ _ | 6 _ _ _ |
. 10 |_ _ _|_ | 3+3 _ _ _|_ |
. 9 |_ _ | | 4+2 _ _ | |
. 8 |_ _|_ _|_ | 2+2+2 _ _|_ _|_ |
. 7 |_ _ _ | | 5+1 _ _ _ | |
. 6 |_ _ _|_ | | 3+2+1 _ _ _|_ | |
. 5 |_ _ | | | 4+1+1 _ _ | | |
. 4 |_ _|_ | | | 2+2+1+1 _ _|_ | | |
. 3 |_ _ | | | | 3+1+1+1 _ _ | | | |
. 2 |_ | | | | | 2+1+1+1+1 _ | | | | |
. 1 |_|_|_|_|_|_| 1+1+1+1+1+1 | | | | | |
.
Then we use the elements from the above diagram to draw an infinite Dyck path in which the j-th odd-indexed segment has A141285(j) up-steps and the j-th even-indexed segment has A194446(j) down-steps.
For the illustration of initial terms we use two opposite Dyck paths, as shown below:
11 ...........................................................
. /\
. /
. /
7 .................................. /
. /\ /
5 .................... / \ /\/
. /\ / \ /\ /
3 .......... / \ / \ / \/
2 ..... /\ / \ /\/ \ /
1 .. /\ / \ /\/ \ / \ /\/
0 /\/ \/ \/ \/ \/
. \/\ /\ /\ /\ /\
. \/ \ / \/\ / \ / \/\
. 1 \/ \ / \/\ / \
. 4 \ / \ / \ /\
. 9 \/ \ / \/ \
. \ / \/\
. 28 \/ \
. \
. 54 \
. \
. \/
.
The diagram is infinite. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n).
Calculations:
a(1) = 1.
a(2) = 2^2 = 4.
a(3) = 3^2 = 9.
a(4) = 2^2-1^2+5^2 = 4-1+25 = 28.
a(5) = 3^2-2^2+7^2 = 9-4+49 = 54.
a(6) = 2^2-1^2+5^2-3^2+6^2-5^2+11^2 = 4-1+25-9+36-25+121 = 151.
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