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This sequence on OEIS gives the Number of Dyck n-paths with at least one UUU.

QUESTIONS. Is there a formula for the Number of Dyck n-paths with at least one $$UUUU...U \qquad \text{$k$ U's?}$$ Maybe we can start with small k=4,5,...? The result for k=3 seems to be very nice. Recall that a dyck path is a lattice path of length 2n with steps U(ups corresponding to (1,1)) and D(downs corresponding to (1,-1)) such that it starts at (0,0) and never goes below the x-axis.

Recall that a dyck $n$-path is a lattice path of length $2n$ with steps $U$ (ups corresponding to $(1,1)$) and $D$ (downs corresponding to $(1,-1)$) such that it starts at $(0,0)$ and never goes below the $x$-axis.

This sequence on OEIS gives the Number of Dyck $n$-paths with at least one $UUU$.

QUESTIONS. Is there a formula for the Number of Dyck n-paths with at least one $$UUUU...U \qquad \text{$k$ U's?}$$ Maybe we can start with small $k=4,5,\dots$? The result for $k=3$ seems to be very nice.

This sequence on OEIS gives the Number of Dyck n-paths with at least one UUU.

QUESTIONS. Is there a formula for the Number of Dyck n-paths with at least one $$UUUU...U \qquad \text{$n$ U's?}$$ Maybe we can start with small n=4,5,...? The result for n=3 seems to be very nice. Recall that a dyck path is a lattice path of length 2n with steps U(ups corresponding to (1,1)) and D(downs corresponding to (1,-1)) such that it starts at (0,0) and never goes below the x-axis.

This sequence on OEIS gives the Number of Dyck n-paths with at least one UUU.

QUESTIONS. Is there a formula for the Number of Dyck n-paths with at least one $$UUUU...U \qquad \text{$k$ U's?}$$ Maybe we can start with small k=4,5,...? The result for k=3 seems to be very nice. Recall that a dyck path is a lattice path of length 2n with steps U(ups corresponding to (1,1)) and D(downs corresponding to (1,-1)) such that it starts at (0,0) and never goes below the x-axis.

http://oeis.org/search?q=1%2C5%2C21%2C81%2C302&sort=&language=english&go=Search gives the Number of Dyck n-paths with at least one UUU. Is there a formula for the Number of Dyck n-paths with at least one UUUU...U (k U's)? Maybe we can start with small k=4,5,...? The result for k=3 seems to be very nice. Recall that a dyck path is a lattice path of length 2n with steps U(ups corresponding to (1,1)) and D(downs corresponding to (1,-1)) such that it starts at (0,0) and never goes below the x-axis.

This sequence on OEIS gives the Number of Dyck n-paths with at least one UUU.

QUESTIONS. Is there a formula for the Number of Dyck n-paths with at least one $$UUUU...U \qquad \text{$n$ U's?}$$ Maybe we can start with small n=4,5,...? The result for n=3 seems to be very nice. Recall that a dyck path is a lattice path of length 2n with steps U(ups corresponding to (1,1)) and D(downs corresponding to (1,-1)) such that it starts at (0,0) and never goes below the x-axis.