Fix an integer $t\geq0$ and consider the sequence $T_{0,t}=1$ and for $n\geq1$, with $k*=n-1-k+t\mod n$, $$T_{n,t}=\sum_{k=0}^{n-1}T_{k,t}T_{k*,t}.$$ EXAMPLES. Some initial terms:

(a) the case $t=0$: $1, 1, 2, 5, 14, 42, 132, 429, 1430,\dots$ (restores the Catalan numbers)

(b) the case $t=1$: $1, 1, 2, 5, 15, 51, 188, 731, 2950, \dots$ (results in number of Schroeder paths)

(c) the case $t=2$: $1, 1, 2, 6, 26, 142, 882, 5910, 41610, \dots$ (unavailable on OEIS)

QUESTION. Is there a common approach to offer a combinatorial interpretation of $T_{n,t}$, for all $t$?

Fix an integer $t\geq0$ and consider the sequence $T_{0,t}=1$ and for $n\geq1$, with $k*=n-1-k+t\mod n$, $$T_{n,t}=\sum_{k=0}^{n-1}T_{k,t}T_{k*,t}.$$ EXAMPLES. Some initial terms:

(a) the case $t=0$: $1, 1, 2, 5, 14, 42, 132, 429, 1430,\dots$ (restores the Catalan numbers)

(b) the case $t=1$: $1, 1, 2, 5, 15, 51, 188, 731, 2950, \dots$ (results in number of Schroeder paths)

(c) the case $t=2$: $1, 1, 2, 6, 26, 142, 882, 5910, 41610, \dots$ (unavailable on OEIS)

QUESTION. Is there a common approach for a combinatorial interpretation of $T_{n,t}$, for all $t$?