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 [-1, 0) (13)

• A207607 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 8 · 1, 12, 30, 21 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  [−1, 0) nonnegative
• A116414 → x ↦ x/(x+1)
  1 · 4, 5 · 13, 34, 22 · 40, 162, 216, 95 … row sums 1, 9, 69, 513, 3789, 27945, 206037, …
  [−1, 0) nonnegative
• A078812 → x ↦ x/(x+1)
  1 · 2, 3 · 3, 10, 8 · 4, 22, 38, 21 … row sums 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  [−1, 0) nonnegative
• A033282 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 11 · 1, 12, 42, 45 … row sums 1, 4, 19, 100, 562, 3304, 20071, …
  [−1, 0) nonnegative
• A112493: Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.
  1 · 1, 1 · 1, 4, 3 · 1, 11, 25, 15 … row sums 1, 2, 8, 52, 472, 5504, 78416, 1320064, …
  A112493 [−1, 0) nonnegative
• A028246: Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
  1 · 1, 1 · 1, 3, 2 · 1, 7, 12, 6 … row sums 1, 2, 6, 26, 150, 1082, 9366, 94586, …
  A028246 [−1, 0) nonnegative
• A063007 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 13 · 1, 15, 57, 63 … row sums 1, 4, 22, 136, 886, 5944, 40636, …
  [−1, 0) nonnegative
• A112486: Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.
  1 · 1, 1 · 2, 5, 3 · 6, 26, 35, 15 … row sums 1, 2, 10, 82, 938, 13778, 247210, …
  A112486 [−1, 0) nonnegative
• A207606 → x ↦ x/(x+1)
  4, 15, 13 · 5, 31, 58, 34 · 6, 54, 162, 201, 89 … row sums 32, 128, 512, 2048, 8192, 32768, …
  [−1, 0) nonnegative
• A111125 → x ↦ x/(x+1)
  1 · 3, 4 · 5, 15, 11 · 7, 35, 56, 29 … row sums 1, 7, 31, 127, 511, 2047, 8191, 32767, …
  [−1, 0) nonnegative
• A085478 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 20, 13 … row sums 1, 3, 11, 43, 171, 683, 2731, 10923, …
  [−1, 0) nonnegative
• A110441 → x ↦ x/(x+1)
  1 · 3, 4 · 7, 20, 14 · 15, 68, 100, 48 … row sums 1, 7, 41, 231, 1289, 7175, 39913, …
  [−1, 0) nonnegative
• A123519 → x ↦ x/(x+1)
  1, 8, 11 · 1, 15, 47, 41 · 1, 24, 126, 240, 153 … row sums 20, 104, 544, 2848, 14912, 78080, …
  [−1, 0) nonnegative