  The first author recently introduced an integer sequence now numbered A355519
in OEIS. This sequence arose from counting bracket tournaments; its study
evokes the analysis of the Catalan triangle (sequence A009766 in OEIS) and the
related Catalan numbers, sequence A000108 in OEIS. We therefore introduce a
general construction that places these sequences on the same footing and
suggests many new integer sequences. We provide code for performing this
construction and a lengthy list of examples. This construction determines a
function from input sequences to output sequences. Some of the resulting output
sequences are in OEIS and others are not.
