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, 1] (10)

• basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).
  1 · 0, 2 · -2, 2, 3 · 0, -8, 8, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• A341287: Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).
  A023531 → basis A341287
  1 · 0, 1 · -1, 1, 1 · -1, -2, 3, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A341287 (−∞, 1]
• A181371: Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/2)).
  A011973 → x ↦ (1/9)x−1/9
  1 · 3 · 8, 1 · 21, 6 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A181371 (−∞, 1] nonnegative
• basis-map coefficients of A371080, shifted → x ↦ x−1
  1 · -1, 1 · -2, 1, 1 · -10, 3, 6, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 1] rational roots
• basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 0, 2 · -3, 3, 3 · 0, -12, 12, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• 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
• A118424: Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 001 (n,k>=0).
  A102547 → x ↦ (1/8)x−1/8
  1 · 2 · 4 · 7, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A118424 (−∞, 1] nonnegative
• A118884: Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).
  A318772 → x ↦ (1/48)x−1/48
  1 · 2 · 4 · 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A118884 (−∞, 1] nonnegative
• A261711: Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.
  A011973 → x ↦ (1/16)x−1/16
  1 · 4 · 15, 1 · 56, 8 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A261711 (−∞, 1] nonnegative
• A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.
  A011973 → x ↦ (1/4)x−1/4
  1 · 2 · 3, 1 · 4, 4 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A034867 A202064 (−∞, 1] nonnegative