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%I #22 Nov 21 2014 02:17:26
%S 1,4,9,28,54,151
%N Area between two valleys at height 0 under the infinite Dyck path related to partitions in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1, multiplied by 2.
%e 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:
%e .
%e . j Diagram 1 Partitions Diagram 2
%e . _ _ _ _ _ _ _ _ _ _ _ _
%e . 11 |_ _ _ | 6 _ _ _ |
%e . 10 |_ _ _|_ | 3+3 _ _ _|_ |
%e . 9 |_ _ | | 4+2 _ _ | |
%e . 8 |_ _|_ _|_ | 2+2+2 _ _|_ _|_ |
%e . 7 |_ _ _ | | 5+1 _ _ _ | |
%e . 6 |_ _ _|_ | | 3+2+1 _ _ _|_ | |
%e . 5 |_ _ | | | 4+1+1 _ _ | | |
%e . 4 |_ _|_ | | | 2+2+1+1 _ _|_ | | |
%e . 3 |_ _ | | | | 3+1+1+1 _ _ | | | |
%e . 2 |_ | | | | | 2+1+1+1+1 _ | | | | |
%e . 1 |_|_|_|_|_|_| 1+1+1+1+1+1 | | | | | |
%e .
%e 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.
%e For the illustration of initial terms we use two opposite Dyck paths, as shown below:
%e 11 ...........................................................
%e . /\
%e . /
%e . /
%e 7 .................................. /
%e . /\ /
%e 5 .................... / \ /\/
%e . /\ / \ /\ /
%e 3 .......... / \ / \ / \/
%e 2 ..... /\ / \ /\/ \ /
%e 1 .. /\ / \ /\/ \ / \ /\/
%e 0 /\/ \/ \/ \/ \/
%e . \/\ /\ /\ /\ /\
%e . \/ \ / \/\ / \ / \/\
%e . 1 \/ \ / \/\ / \
%e . 4 \ / \ / \ /\
%e . 9 \/ \ / \/ \
%e . \ / \/\
%e . 28 \/ \
%e . \
%e . 54 \
%e . \
%e . \/
%e .
%e The diagram is infinite. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n).
%e Calculations:
%e a(1) = 1.
%e a(2) = 2^2 = 4.
%e a(3) = 3^2 = 9.
%e a(4) = 2^2-1^2+5^2 = 4-1+25 = 28.
%e a(5) = 3^2-2^2+7^2 = 9-4+49 = 54.
%e a(6) = 2^2-1^2+5^2-3^2+6^2-5^2+11^2 = 4-1+25-9+36-25+121 = 151.
%Y Cf. A000041, A135010, A141285, A193870, A194446, A194447, A206437, A211009, A211978, A220517, A225600, A225610, A228109, A228110, A228350, A230440, A233968.
%K nonn,more
%O 1,2
%A _Omar E. Pol_, Nov 08 2014
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