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)

• A038231 → x ↦ x/(x+1)
  1 · 4, 5 · 16, 40, 25 · 64, 240, 300, 125 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  [−1, 0] nonnegative rational roots
• A064861 → x ↦ x/(x+1)
  1, 5, 6 · 1, 8, 21, 18 · 1, 10, 37, 60, 36 … row sums 12, 48, 144, 576, 1728, 6912, 20736, …
  [−1, 0] nonnegative rational roots
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
  A010054 → reverse
  1 · 0, 3 · 0, 0, 6 · 0, 0, 0, 10 … row sums 1, 3, 6, 10, 15, 21, 28, 36, …
  A127773 [−1, 0] nonnegative rational roots
• A094816 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 11, 25, 16 … row sums 1, 3, 11, 53, 345, 2947, 31411, 400437, …
  [−1, 0] nonnegative
• A081577 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 10, 24, 16 … row sums 1, 3, 13, 51, 205, 819, 3277, 13107, …
  [−1, 0] nonnegative
• A008459 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  [−1, 0] nonnegative
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
  1 · 2, 3 · 4, 12, 9 · 8, 36, 54, 27 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013620 (−∞, 0) [−1, 0] nonnegative rational roots
• A007318 → (1/2+(1/2)n)θ+1+n → x ↦ x/(x+1)
  1 · 2, 5 · 3, 15, 18 · 4, 30, 72, 56 … row sums 1, 7, 36, 162, 675, 2673, 10206, 37908, …
  [−1, 0] nonnegative rational roots
• A137320 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 6 · 0, 4, 20, 24 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  [−1, 0] nonnegative rational roots
• A143495 → x ↦ x/(x+1)
  1 · 3, 4 · 9, 25, 17 · 27, 118, 167, 77 … row sums 1, 7, 51, 389, 3121, 26423, 236403, …
  [−1, 0] nonnegative
• basis-map coefficients of A scaled Hermite triangle.
  0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] nonnegative rational roots
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
  A010054 → D∘x
  1 · 3 · 5 · 7 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  A198954 (−∞, −1] [−1, 0] nonnegative rational roots
• A007318 → D+1 → x ↦ x/(x+1)
  1 · 2, 3 · 3, 10, 8 · 4, 21, 36, 20 … row sums 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  [−1, 0] nonnegative rational roots
• A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A033282 (−1, 0) nonnegative
• A026374 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 7, 15, 10 … row sums 1, 3, 11, 33, 121, 363, 1331, 3993, …
  [−1, 0] nonnegative
• A081578 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 12, 30, 20 … row sums 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  [−1, 0] nonnegative
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
  1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A013610 (−∞, 0) [−1, 0] nonnegative rational roots
• A091044 → ⊙ C(n+2,k+1) → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 17, 45, 30 … row sums 1, 3, 15, 93, 653, 4983, 40339, 341337, …
  [−1, 0] nonnegative
• A099097 → x ↦ x/(x+1)
  1 · 0, 3 · 0, 1, 10 · 0, 0, 6, 33 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  [−1, 0] nonnegative
• A028326 → x ↦ x/(x+1)
  2 · 2, 4 · 2, 8, 8 · 2, 12, 24, 16 … row sums 2, 6, 18, 54, 162, 486, 1458, 4374, …
  [−1, 0] nonnegative rational roots
• A082985 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 7, 11 · 1, 10, 31, 29 … row sums 1, 5, 19, 71, 265, 989, 3691, 13775, …
  [−1, 0] nonnegative
• 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
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
  1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A013609 (−∞, 0) [−1, 0] nonnegative rational roots
• A120910 → x ↦ x/(x+1)
  3 · 6, 9 · 12, 36, 27 · 24, 108, 162, 81 … row sums 3, 15, 75, 375, 1875, 9375, 46875, …
  [−1, 0] nonnegative rational roots
• A133087 → x ↦ x/(x+1)
  1 · 2, 3 · 1, 4, 4 · 2, 11, 20, 12 … row sums 1, 5, 9, 45, 81, 405, 729, 3645, …
  [−1, 0] nonnegative rational roots
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
  1 · -1, -2 · 1, 4, 4 · -1, -6, -12, -8 … row sums 1, -3, 9, -27, 81, -243, 729, -2187, …
  [−1, 0] rational roots
• A126351 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 12, 40, 37 … row sums 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  [−1, 0] nonnegative
• A035607 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 7 · 1, 9, 23, 17 … row sums 1, 4, 14, 50, 178, 634, 2258, 8042, …
  [−1, 0] nonnegative
• A023531 → 2D+1+2n → x ↦ x/(x+1)
  1 · 2, 5 · 0, 4, 9 · 0, 0, 6, 13 … row sums 1, 7, 13, 19, 25, 31, 37, 43, …
  [−1, 0] nonnegative rational roots
• A113413 → x ↦ x/(x+1)
  1 · 2, 3 · 2, 8, 7 · 2, 14, 28, 17 … row sums 1, 5, 17, 61, 217, 773, 2753, 9805, …
  [−1, 0] nonnegative
• A039756 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 12, 34, 24 … row sums 1, 3, 13, 71, 457, 3355, 27509, 248127, …
  [−1, 0] nonnegative
• A122431 → x ↦ x/(x+1)
  1, 6, 12, 8 · 0, 1, 6, 12, 8 · 0, 0, 1, 6, 12, 8 … row sums 27, 27, 27, 27, 27, 27, 27, 27, …
  [−1, 0] nonnegative rational roots
• A088996: Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.
  1 · 0, 1 · 0, 1, 2 · 0, 2, 7, 6 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A088996 [−1, 0] nonnegative rational roots
• A102547 → reverse → x ↦ x/(x+1)
  0, 1 · 0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 1, 2 … row sums 1, 1, 1, 3, 5, 7, 13, 23, …
  [−1, 0] nonnegative
• A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n² → x ↦ x/(x+1)
  1 · 3, 7 · 6, 28, 32 · 10, 70, 160, 120 … row sums 1, 10, 66, 360, 1755, 7938, 34020, …
  [−1, 0] nonnegative rational roots
• A159854 → x ↦ x/(x+1)
  0, 0, 1 · 0, 0, 1, 2 · 0, 0, 1, 4, 4 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  [−1, 0] nonnegative rational roots
• A007318 → (θ+1)² → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 18 · 1, 15, 54, 56 … row sums 1, 6, 29, 126, 513, 1998, 7533, 27702, …
  [−1, 0] nonnegative
• A039755 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 16, 38, 24 … row sums 1, 3, 13, 79, 601, 5339, 53861, 607527, …
  [−1, 0] nonnegative
• A085880 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 8, 8 · 5, 30, 60, 40 … row sums 1, 3, 18, 135, 1134, 10206, 96228, …
  [−1, 0] nonnegative rational roots
• A038208 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 16, 16 · 8, 48, 96, 64 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  [−1, 0] nonnegative rational roots
• basis-map coefficients of A026729 → basis A056242 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 12, 37, 34 … row sums 1, 4, 18, 84, 396, 1872, 8856, 41904, …
  [−1, 0] nonnegative
• A106800 → reverse (shift +1) → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 5, 5 … row sums 1, 1, 3, 11, 49, 257, 1539, 10299, …
  [−1, 0] nonnegative
• A091044 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 16, 16 · 4, 40, 96, 64 … row sums 1, 6, 35, 204, 1189, 6930, 40391, …
  [−1, 0] nonnegative
• A114687 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 11 · 1, 15, 51, 45 … row sums 1, 4, 20, 112, 672, 4224, 27456, …
  [−1, 0] nonnegative
• A062264 → x ↦ x/(x+1)
  1 · 1, 6 · 1, 14, 28 · 1, 24, 108, 120 … row sums 1, 7, 43, 253, 1462, 8378, 47818, …
  [−1, 0] nonnegative
• A249250 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 8, 10 · 1, 12, 36, 28 … row sums 1, 5, 19, 77, 307, 1229, 4915, 19661, …
  [−1, 0] nonnegative
• A116412 → x ↦ x/(x+1)
  1 · 3, 4 · 6, 18, 13 · 12, 57, 87, 43 … row sums 1, 7, 37, 199, 1069, 5743, 30853, …
  [−1, 0] nonnegative
• A007318 → 2θ+1 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 8, 12 · 1, 12, 36, 32 … row sums 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  [−1, 0] nonnegative rational roots
• A143685 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 13, 13 · 1, 24, 66, 44 … row sums 1, 3, 27, 135, 891, 5103, 31347, …
  [−1, 0] nonnegative
• A007318 → ⊙ C(n+1,k+1) (shift +2) → x ↦ x/(x+1)
  8 · 12, 30 · 16, 96, 112 · 20, 210, 560, 420 … row sums 8, 42, 224, 1210, 6600, 36274, 200576, …
  [−1, 0] nonnegative