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 [0, +inf) (6)

• A039757: Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).
  A028338 → x ↦ −x (shift -1)
  1 · -1, 1 · 3, -4, 1 · -15, 23, -9, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  A039757 [0, ∞) rational roots
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] [0, ∞) nonnegative rational roots
• A021010: Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).
  A144084 → x ↦ −x (shift -1)
  1 · -1, 1 · 1, -4, 2 · -1, 9, -18, 6 … row sums 1, 0, -1, -4, -15, -56, -185, -204, …
  A021010 [0, ∞)
• A023531 → x ↦ x−1
  1 · -1, 1 · 1, -2, 1 · -1, 3, -3, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 1] [0, ∞) rational roots
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A023531 [−1, 0] [0, ∞) nonnegative rational roots
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0
  1 · 0, 0, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] [0, ∞) nonnegative rational roots