Root regions
Nodes grouped by the interval that contains every root of every row (proved, for all n). A node can have several proved regions; the sharpest is the smallest interval.
Nodes with all roots in (-2, 0) (6)
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chebyshev_t → x ↦ x+1
1 · 1, 1 · 1, 4, 2 · 1, 9, 12, 4 … row sums 1, 2, 7, 26, 97, 362, 1351, 5042, …
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A225476: Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
1 · 1, 1 · 1, 4, 2 · 1, 13, 18, 6 … row sums 1, 2, 7, 38, 277, 2522, 27547, 351038, …
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Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).
1, 6, 4 · 1, 12, 20, 8 · 1, 20, 60, 56, 16 … row sums 11, 41, 153, 571, 2131, 7953, 29681, …
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chebyshev_u → x ↦ x+1
1 · 2, 2 · 3, 8, 4 · 4, 20, 24, 8 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
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A053117 → x ↦ x+1
1 · 2, 2 · 3, 8, 4 · 4, 20, 24, 8 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
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A358735: Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.
1 · 1, 1 · 1, 4, 2 · 1, 10, 16, 6 … row sums 1, 2, 7, 33, 191, 1304, 10241, 90865, …