A111577
identification: the entry's rule defines the sequence for every index; tail: the sequence is the entry's tail after some prefix rows; match_*: finite agreement only.
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A111577: Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.
(identification, match triangle)
1 · 1, 1 · 1, 5, 1 · 1, 21, 12, 1 … row sums 1, 2, 7, 35, 214, 1523, 12349, 112052, …