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)

• A091042 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 12, 16 · 1, 24, 80, 64 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  [−1, 0] nonnegative
• A113953 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 2, 3 · 0, 0, 4, 5 … row sums 1, 1, 5, 9, 29, 65, 181, 441, …
  [−1, 0] nonnegative
• A272866 → x ↦ x/(x+1)
  1 · 1, 5, 5 · 1, 10, 35, 50, 25 … row sums 1, 11, 121, 1331, 14641, 161051, …
  [−1, 0] nonnegative
• A099040 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 6 · 0, 0, 8, 16 … row sums 1, 2, 8, 24, 80, 256, 832, 2688, …
  [−1, 0] nonnegative
• A081579 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 8, 8 · 1, 14, 36, 24 … row sums 1, 3, 17, 75, 361, 1683, 7937, 37275, …
  [−1, 0] nonnegative
• delannoy → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 8, 18, 12 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  [−1, 0] nonnegative
• A249251 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 8, 13 · 1, 12, 42, 43 … row sums 1, 5, 22, 98, 436, 1940, 8632, 38408, …
  [−1, 0] nonnegative
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
  1, 4, 4 · 3, 18, 36, 24 · 6, 48, 144, 192, 96 … row sums 0, 0, 9, 81, 486, 2430, 10935, 45927, …
  [−1, 0] nonnegative rational roots
• A023531 → (1/2)D²+2D+1 → x ↦ x/(x+1)
  1 · 2, 3 · 1, 6, 6 · 0, 3, 12, 10 … row sums 1, 5, 13, 25, 41, 61, 85, 113, …
  [−1, 0] nonnegative
• A178820 → x ↦ x/(x+1)
  1 · 4, 8 · 10, 40, 40 · 20, 120, 240, 160 … row sums 1, 12, 90, 540, 2835, 13608, 61236, …
  [−1, 0] nonnegative rational roots
• A008278 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 22, 15 … row sums 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  [−1, 0] nonnegative
• A145597 → x ↦ x/(x+1)
  1 · 3, 6 · 6, 27, 27 · 10, 75, 165, 110 … row sums 1, 9, 60, 360, 2061, 11529, 63728, …
  [−1, 0] nonnegative
• A143494 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 13, 10 · 8, 43, 71, 37 … row sums 1, 5, 27, 159, 1025, 7221, 55307, …
  [−1, 0] nonnegative
• A027465 → x ↦ x/(x+1)
  1 · 3, 4 · 9, 24, 16 · 27, 108, 144, 64 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  [−1, 0] nonnegative 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
• A008288 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 8, 18, 12 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  [−1, 0] nonnegative
• A102662 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 7, 9 · 1, 10, 28, 22 … row sums 1, 5, 17, 61, 217, 773, 2753, 9805, …
  [−1, 0] nonnegative
• A059473 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 18, 18 · 8, 56, 120, 80 … row sums 1, 6, 40, 264, 1744, 11520, 76096, …
  [−1, 0] nonnegative
• A062145 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 12, 21 · 1, 21, 84, 84 … row sums 1, 6, 34, 190, 1059, 5908, 33028, …
  [−1, 0] nonnegative
• A086645 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 8, 8 · 1, 18, 48, 32 … row sums 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  [−1, 0] nonnegative
• A062190 → x ↦ x/(x+1)
  1 · 1, 7 · 1, 16, 36 · 1, 27, 135, 165 … row sums 1, 8, 53, 328, 1966, 11584, 67582, …
  [−1, 0] nonnegative
• A143683 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 12, 12 · 1, 22, 60, 40 … row sums 1, 3, 25, 123, 769, 4275, 25129, …
  [−1, 0] nonnegative
• A075497 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 14, 11 · 8, 52, 92, 49 … row sums 1, 5, 29, 201, 1657, 15821, 170389, …
  [−1, 0] nonnegative
• A227450 → x ↦ x/(x+1)
  2 · 2, 6 · 2, 16, 22 · 2, 34, 110, 94 … row sums 2, 8, 40, 240, 1664, 12992, 112256, …
  [−1, 0] nonnegative
• A056242: Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).
  basis-map coefficients of A026729 → basis A056242
  1 · 1, 2 · 1, 5, 4 · 1, 9, 16, 8 … row sums 1, 3, 10, 34, 116, 396, 1352, 4616, …
  A056242 [−1, 0] nonnegative
• A008277 → θ+2 (shift +1) → x ↦ x/(x+1)
  2 · 2, 5 · 2, 13, 15 · 2, 27, 72, 52 … row sums 2, 7, 30, 153, 898, 5919, 43102, …
  [−1, 0] nonnegative
• A008277 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 10 · 1, 17, 49, 37 … row sums 1, 4, 19, 104, 641, 4380, 32803, …
  [−1, 0] nonnegative
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
  1 · 1, 4 · 1, 8, 16 · 1, 12, 48, 64 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013611 (−∞, 0) [−1, 0] nonnegative rational roots
• A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
  1 · 1, 2 · 1, 5, 6 · 1, 9, 26, 24 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  A145324 (−∞, 0) [−1, 0] nonnegative rational roots
• A344565 → x ↦ x/(x+1)
  3 · 15, 30 · 45, 180, 180 · 105, 630, 1260, 840 … row sums 3, 45, 405, 2835, 17010, 91854, 459270, …
  [−1, 0] nonnegative rational roots
• A108838 → ⊙ C(n+1,k+1) (shift +1) → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 13, 33, 22 … row sums 1, 3, 13, 69, 417, 2763, 19609, 146793, …
  [−1, 0] nonnegative
• A121574 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 13, 10 · 8, 40, 64, 33 … row sums 1, 5, 27, 145, 779, 4185, 22483, …
  [−1, 0] nonnegative
• A030528 → x ↦ x/(x+1)
  1 · 1, 2 · 0, 2, 3 · 0, 1, 5, 5 … row sums 1, 3, 5, 11, 21, 43, 85, 171, …
  [−1, 0] nonnegative
• A249248 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 9, 11 · 1, 15, 49, 38 … row sums 1, 5, 21, 103, 519, 2793, 15645, 91623, …
  [−1, 0] nonnegative
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A010054 (−∞, −1] [−1, 0] [−3−2√2, −3+2√2] nonnegative rational roots
• A154986 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … row sums 1, 3, 13, 75, 537, 4611, 46053, 525483, …
  [−1, 0] nonnegative
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
  1 · 3, 6 · 6, 24, 24 · 10, 60, 120, 80 … row sums 1, 9, 54, 270, 1215, 5103, 20412, …
  [−1, 0] nonnegative 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. shifted by x -> -x + 0
  1 · 0, -1 · 0, 0, 1 · 0, 0, 0, -1 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  [−1, 0] rational roots
• A104684 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 20, 90, 132, 63 … row sums 1, 5, 37, 305, 2641, 23525, 213445, …
  [−1, 0] nonnegative
• A131689 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 3 · 0, 1, 8, 13 … row sums 1, 1, 4, 22, 160, 1456, 15904, 202672, …
  [−1, 0] nonnegative
• A114656 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 14, 11 · 8, 48, 84, 45 … row sums 1, 5, 29, 185, 1257, 8925, 65445, …
  [−1, 0] nonnegative
• A120909 → x ↦ x/(x+1)
  3 · 3, 9 · 3, 18, 27 · 3, 27, 81, 81 … row sums 3, 12, 48, 192, 768, 3072, 12288, …
  [−1, 0] nonnegative rational roots
• A007318 → basis A049020 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 15, 67, 98, 47 … row sums 1, 5, 31, 227, 1897, 17693, 181351, …
  [−1, 0] nonnegative
• A172431 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 8 · 1, 9, 25, 21 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  [−1, 0] nonnegative
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
  1 · 0, -1 · 0, 0, 1 · 0, 0, 0, -1 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  [−1, 0] rational roots
• A172431 → ⊙ C(n,k+1) (shift -1) → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 12, 31, 21 … row sums 1, 3, 13, 65, 353, 2023, 12041, 73713, …
  [−1, 0] nonnegative
• A114655 → x ↦ x/(x+1)
  2 · 4, 6 · 8, 28, 22 · 16, 96, 168, 90 … row sums 2, 10, 58, 370, 2514, 17850, 130890, …
  [−1, 0] nonnegative
• A122542 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 2, 3 · 0, 2, 8, 7 … row sums 1, 1, 5, 17, 61, 217, 773, 2753, …
  [−1, 0] nonnegative
• narayana → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  [−1, 0] nonnegative
• A130534 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 6, 29, 46, 24 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  [−1, 0] nonnegative rational roots