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] (147)

• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
  1 · 4, 8 · 9, 36, 36 · 16, 96, 192, 128 … row sums 1, 12, 81, 432, 2025, 8748, 35721, …
  [−1, 0] nonnegative rational roots
• A054142 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 8, 19, 13 … row sums 1, 3, 11, 41, 153, 571, 2131, 7953, …
  [−1, 0] nonnegative
• A046716 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 23, 16 … row sums 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  [−1, 0] nonnegative
• A081581 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 10, 10 · 1, 18, 48, 32 … row sums 1, 3, 21, 99, 549, 2835, 15093, 79299, …
  [−1, 0] nonnegative
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).
  1 · 0, 2 · 0, 0, 4 · 0, 0, 0, 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A130123 [−1, 0] nonnegative rational roots
• A007318 → D → x ↦ x/(x+1)
  1 · 2, 4 · 3, 12, 12 · 4, 24, 48, 32 … row sums 1, 6, 27, 108, 405, 1458, 5103, 17496, …
  [−1, 0] nonnegative rational roots
• A132812 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 15, 15 · 4, 36, 84, 56 … row sums 1, 6, 33, 180, 985, 5418, 29953, …
  [−1, 0] nonnegative
• A100862 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … row sums 1, 3, 11, 45, 201, 963, 4899, 26253, …
  [−1, 0] nonnegative
• A103633 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 0, 1, 2 … row sums 1, 1, 3, 3, 9, 9, 27, 27, …
  [−1, 0] nonnegative rational roots
• A013611 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 25 · 1, 15, 75, 125 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  [−1, 0] nonnegative rational roots
• A023531 → basis A049020 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 5, 25, 41, 22 … row sums 1, 3, 15, 93, 681, 5691, 53079, 544053, …
  [−1, 0] nonnegative
• A152842 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 6, 8 · 1, 10, 32, 32 … row sums 1, 5, 15, 75, 225, 1125, 3375, 16875, …
  [−1, 0] nonnegative rational roots
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
  1 · 3, 5 · 9, 30, 25 · 27, 135, 225, 125 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A013622 (−∞, 0) [−1, 0] nonnegative rational roots
• A391533 → x ↦ x/(x+1)
  4, 12, 9 · 4, 20, 33, 18 · 4, 28, 73, 84, 36 … row sums 25, 75, 225, 675, 2025, 6075, 18225, …
  [−1, 0] nonnegative rational roots
• A008277 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 10, 23, 15 … row sums 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  [−1, 0] nonnegative
• A173018 → reverse (shift +1) → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 6, 6 … row sums 1, 1, 3, 13, 75, 541, 4683, 47293, …
  [−1, 0] nonnegative
• A131431: 3n + 1 preceded by n zeros.
  A010054 → reverse
  1 · 0, 4 · 0, 0, 7 · 0, 0, 0, 10 … row sums 1, 4, 7, 10, 13, 16, 19, 22, …
  A131431 [−1, 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
• A081582 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 11, 11 · 1, 20, 54, 36 … row sums 1, 3, 23, 111, 655, 3519, 19727, …
  [−1, 0] nonnegative
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
  1 · 0, -1 · 0, 0, 2 · 0, 0, 0, -6 … row sums 1, -1, 2, -6, 24, -120, 720, -5040, …
  [−1, 0] rational roots
• A145596 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 14, 14 · 4, 32, 72, 48 … row sums 1, 6, 31, 156, 785, 3978, 20335, …
  [−1, 0] nonnegative
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
  A023531 → (θ+1)²
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … row sums 1, 4, 9, 16, 25, 36, 49, 64, …
  A127733 [−1, 0] nonnegative rational roots
• A034870 → x ↦ x/(x+1)
  1 · 1, 4, 4 · 1, 8, 24, 32, 16 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  [−1, 0] nonnegative rational roots
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
  A010054 → reverse
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  A127648 [−1, 0] nonnegative rational roots
• A008278 → x ↦ x+1 (shift +1) → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 15, 68, 101, 49 … row sums 1, 5, 31, 233, 2047, 20417, 226711, …
  [−1, 0] nonnegative
• A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  A019538 (−1, 0) nonnegative
• A103327 → x ↦ x/(x+1)
  1 · 3, 4 · 5, 20, 16 · 7, 56, 112, 64 … row sums 1, 7, 41, 239, 1393, 8119, 47321, …
  [−1, 0] nonnegative
• A026729 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 0, 2, 3 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  [−1, 0] nonnegative
• A232535 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 10, 12 · 1, 21, 64, 48 … row sums 1, 4, 23, 134, 781, 4552, 26531, …
  [−1, 0] nonnegative
• A133080 → x ↦ x/(x+1)
  1 · 1, 2 · 0, 0, 1 · 0, 0, 1, 2 … row sums 1, 3, 1, 3, 1, 3, 1, 3, …
  [−1, 0] nonnegative rational roots
• A113214 → x ↦ x/(x+1)
  1 · 2, 3 · 0, 3, 4 · 0, 2, 8, 7 … row sums 1, 5, 7, 17, 31, 65, 127, 257, …
  [−1, 0] nonnegative
• A135838 → x ↦ x/(x+1)
  1 · 2, 4 · 2, 8, 8 · 4, 24, 48, 32 … row sums 1, 6, 18, 108, 324, 1944, 5832, 34992, …
  [−1, 0] nonnegative rational roots
• A100861 → reverse → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 0, 3, 4 … row sums 1, 1, 3, 7, 25, 81, 331, 1303, …
  [−1, 0] nonnegative
• A178822 → x ↦ x/(x+1)
  1 · 6, 12 · 21, 84, 84 · 56, 336, 672, 448 … row sums 1, 18, 189, 1512, 10206, 61236, 336798, …
  [−1, 0] nonnegative rational roots
• A013620 → x ↦ x/(x+1)
  1 · 2, 5 · 4, 20, 25 · 8, 60, 150, 125 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  [−1, 0] nonnegative rational roots
• A063967 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 3, 16, 28, 16 … row sums 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  [−1, 0] nonnegative
• A007318 → θ+1+n → x ↦ x/(x+1)
  1 · 2, 5 · 3, 14, 16 · 4, 27, 60, 44 … row sums 1, 7, 33, 135, 513, 1863, 6561, 22599, …
  [−1, 0] nonnegative rational roots
• A132774: A natural number operator.
  A023531 → 2D+1+2n
  1 · 2, 3 · 0, 4, 5 · 0, 0, 6, 7 … row sums 1, 5, 9, 13, 17, 21, 25, 29, …
  A132774 [−1, 0] nonnegative rational roots
• A007318 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 8 · 1, 9, 24, 20 … row sums 1, 4, 15, 54, 189, 648, 2187, 7290, …
  [−1, 0] nonnegative rational roots
• A081580 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 9, 9 · 1, 16, 42, 28 … row sums 1, 3, 19, 87, 451, 2223, 11179, 55767, …
  [−1, 0] nonnegative
• A108838 → x ↦ x/(x+1)
  2 · 3, 5 · 4, 16, 14 · 5, 35, 70, 42 … row sums 2, 8, 34, 152, 706, 3376, 16514, 82256, …
  [−1, 0] nonnegative
• A249252 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 17 · 1, 15, 59, 61 … row sums 1, 6, 28, 136, 656, 3168, 15296, 73856, …
  [−1, 0] nonnegative
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
  1 · 0, 1 · 0, 0, 2 · 0, 0, 0, 6 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  A136572 [−1, 0] nonnegative rational roots
• A178821 → x ↦ x/(x+1)
  1 · 5, 10 · 15, 60, 60 · 35, 210, 420, 280 … row sums 1, 15, 135, 945, 5670, 30618, 153090, …
  [−1, 0] nonnegative rational roots
• A062196 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 10, 15 · 1, 18, 63, 56 … row sums 1, 5, 26, 138, 743, 4043, 22180, …
  [−1, 0] nonnegative
• A099093 → x ↦ x/(x+1)
  1 · 0, 3 · 0, 3, 12 · 0, 0, 18, 45 … row sums 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  [−1, 0] nonnegative
• A126216 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 14, 63, 93, 45 … row sums 1, 5, 31, 215, 1597, 12425, 99955, …
  [−1, 0] nonnegative