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 (-inf, 0) (258)

• A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
  4, 4, 1 · 4, 8, 5, 1 · 4, 12, 13, 6, 1 … row sums 9, 18, 36, 72, 144, 288, 576, 1152, …
  A391533 (−∞, 0) nonnegative rational roots
• A086645: Triangle read by rows: T(n, k) = binomial(2n, 2k).
  1 · 1, 1 · 1, 6, 1 · 1, 15, 15, 1 … row sums 1, 2, 8, 32, 128, 512, 2048, 8192, …
  A086645 (−∞, 0) nonnegative
• A257618: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.
  1 · 2, 2 · 4, 40, 4 · 8, 472, 472, 8 … row sums 1, 4, 48, 960, 26880, 967680, 42577920, …
  A257618 (−∞, 0) nonnegative
• Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).
  49, 21, 1 · 343, 343, 42, 1 · 2401, 5145, 1225, 70, 1 … row sums 71, 729, 8842, 125399, 2026249, …
  (−∞, 0) nonnegative
• A069466: Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.
  1 · 2, 2 · 6, 24, 6 · 20, 180, 180, 20 … row sums 1, 4, 36, 400, 4900, 63504, 853776, …
  A069466 (−∞, 0) nonnegative
• A027467: Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).
  1 · 15, 1 · 225, 30, 1 · 3375, 675, 45, 1 … row sums 1, 16, 256, 4096, 65536, 1048576, …
  A027467 (−∞, 0) nonnegative rational roots
• A008292: Eulerian
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A008292 (−∞, 0) nonnegative
• A327997: Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).
  1 · 3, 1 · 9, 7, 1 · 27, 38, 12, 1 … row sums 1, 4, 17, 78, 393, 2208, 13977, 100026, …
  A327997 (−∞, 0) nonnegative