A244968 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.
1, 4, 9, 28, 54, 151
Offset: 1
Examples
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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