Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A120910 → x ↦ x/(x+1)
  3 · 6, 9 · 12, 36, 27 · 24, 108, 162, 81 … Σ 3, 15, 75, 375, 1875, 9375, 46875, …
  [−1, 0]nonnegativerational roots
• A038327 → x ↦ x+1 → D+1
  1 · 14, 1 · 195, 28, 1 · 2704, 585, 42, 1 … Σ 1, 15, 224, 3332, 49392, 729904, …
  (−∞, 0]nonnegativerational roots
• A008277 → θ+1 → reverse
  1 · 2, 1 · 3, 6, 1 · 4, 18, 14, 1 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0]nonnegative
• A178820 → basis A013609
  1 · 8, 8 · 40, 80, 40 · 160, 480, 480, 160 … Σ 1, 16, 160, 1280, 8960, 57344, 344064, …
  (−∞, 0]nonnegativerational roots
• A038284 → basis A007318
  196, 168, 36 · 2744, 3528, 1512, 216 … Σ 400, 8000, 160000, 3200000, 64000000, …
  (−∞, −1]nonnegativerational roots
• A038326 → D+1
  1 · 23, 12 · 385, 552, 144 · 5687, 13860, 9936, 1728 … Σ 1, 35, 1081, 31211, 863857, 23226803, …
  (−∞, 0]nonnegativerational roots
• A131431 → θ+1 → D+1
  1 · 8, 8 · 0, 42, 21 · 0, 0, 120, 40 … Σ 1, 16, 63, 160, 325, 576, 931, 1408, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A112227 → (θ+1)² → θ+1
  0, 0, 27 · 0, 0, 0, 64 · 0, 0, 0, 0, 125 … Σ 27, 64, 125, 216, 343, 512, 729, 1000, …
  (−∞, 0]nonnegativerational roots
• A356546 → basis A013611 → θ+1
  1 · 4, 16 · 24, 192, 288 · 160, 1920, 5760, 5120 … Σ 1, 20, 504, 12960, 332640, 8491392, …
  (−∞, 0]nonnegativerational roots
• A038319 → basis A013609 → D
  320, 200 · 7680, 9600, 3000 … Σ 520, 20280, 703040, 22848800, …
  (−∞, 0]nonnegativerational roots
• A013612 → basis A013610 → x ↦ x/(x+1)
  36, 252, 441 · 216, 2268, 7938, 9261 … Σ 729, 19683, 531441, 14348907, …
  [−1, 0]nonnegativerational roots
• A163936 → reverse → θ+1
  1 · 0, 2 · 0, 2, 6 · 0, 2, 24, 24 … Σ 1, 2, 8, 50, 420, 4410, 55440, 810810, …
  (−∞, 0]nonnegative
• A008278 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 21, 4 … Σ 1, 3, 10, 38, 161, 747, 3753, 20253, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → Veronese 2 (j=1) → D
  -1 · 4 · -10, -2 · 20, 12 … Σ 0, 0, 0, -1, 4, -12, 32, -80, …
  (−∞, 0]
• A104684 → basis A013609 → (θ+1)²
  1 · 3, 8 · 13, 64, 36 · 63, 456, 540, 128 … Σ 1, 11, 113, 1187, 12481, 130075, …
  (−∞, 0]nonnegative
• A209599: Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the op…
  1 · 2 · 3, 1 · 5, 3 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1]nonnegative
• A133214 → basis A007318 → basis A007318
  33, 24, 4 · 245, 258, 84, 8 … Σ 61, 595, 6145, 65527, 712909, 7863667, …
  (−∞, −1]nonnegative
• A100861 → basis A013609 → basis A007318
  1 · 1 · 4, 2 · 10, 6 … Σ 1, 1, 6, 16, 106, 426, 3076, 15856, …
  (−∞, −1]nonnegative
• A232535 → basis A007318 → (θ+1)²
  1 · 3, 8 · 12, 56, 27 · 48, 320, 333, 64 … Σ 1, 11, 95, 765, 5861, 43319, 311603, …
  (−∞, 0]nonnegative
• A188440 → (θ+1)²
  1 · 1 · 1, 8 · 1, 8 … Σ 1, 1, 9, 9, 53, 53, 261, 261, …
  (−∞, 0]nonnegative
• A113214 → basis A013610 → D+1
  1 · 6, 3 · 19, 33, 9 · 46, 165, 144, 27 … Σ 1, 9, 61, 382, 2264, 12960, 72368, …
  (−∞, 0]nonnegative
• A059473 → x ↦ x+1 → θ+1
  1 · 4, 4 · 18, 36, 12 · 80, 240, 168, 32 … Σ 1, 8, 66, 520, 3992, 30048, 222784, …
  (−∞, 0]nonnegative
• A131436 → (θ+1)²
  1 · 0, 12 · 0, 0, 63 · 0, 0, 0, 240 … Σ 1, 12, 63, 240, 775, 2268, 6223, 16320, …
  (−∞, 0]nonnegativerational roots
• A008277 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 22, 15 … Σ 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  (−∞, 0]nonnegative
• A086645 → basis A013609 → D+1
  1 · 4, 2 · 24, 24, 4 · 128, 240, 96, 8 … Σ 1, 6, 52, 472, 4176, 36064, 305728, …
  (−∞, 0]nonnegative
• A099093 → reverse → θ+1
  1 · 3 · 9, 6 · 27, 36 … Σ 1, 3, 15, 63, 270, 1134, 4725, 19521, …
  (−∞, 0]nonnegative
• A027465 → basis A008277 → θ+1
  1 · 4, 2 · 16, 18, 3 · 64, 122, 45, 4 … Σ 1, 6, 37, 235, 1540, 10427, 73013, …
  (−∞, 0]nonnegative
• chebyshev_u → D
  2 · 0, 8 · -4, 0, 24 · 0, -24, 0, 64 … Σ 0, 2, 8, 20, 40, 70, 112, 168, …
  (−∞, ∞)
• A038331 → (θ+1)²
  144, 480, 225 · 1728, 8640, 8100, 2000 … Σ 849, 20468, 465001, 10140432, …
  (−∞, 0]nonnegative
• A126216 → D → θ+1
  1 · 5, 4 · 21, 36, 9 · 84, 224, 126, 16 … Σ 0, 1, 9, 66, 450, 2955, 18963, 119812, …
  (−∞, 0]nonnegative
• A133087 → x ↦ x/(x+1)
  1 · 2, 3 · 1, 4, 4 · 2, 11, 20, 12 … Σ 1, 5, 9, 45, 81, 405, 729, 3645, …
  [−1, 0]nonnegativerational roots
• A030528 → basis A013609 → basis A007318
  1 · 4, 2 · 15, 16, 4 · 57, 92, 48, 8 … Σ 1, 6, 35, 205, 1200, 7025, 41125, …
  (−∞, −1]nonnegative
• A143683 → D
  1 · 10, 2 · 19, 38, 3 · 28, 236, 84, 4 … Σ 0, 1, 12, 60, 352, 1680, 8256, 38080, …
  (−∞, 0]nonnegative
• A257624 → D+1
  1 · 10, 5 · 105, 130, 25 · 1040, 2745, 1290, 125 … Σ 1, 15, 260, 5200, 118560, 3043040, …
  (−∞, 0]nonnegative
• A334218 → (θ+1)²
  1 · 1 · 0, 8 · 0, 12, 27 … Σ 1, 1, 8, 39, 224, 1520, 11880, 105000, …
  (−∞, 0]nonnegative
• A075499 → D+1
  28, 14, 1 · 176, 160, 27, 1 · 1216, 1760, 520, 44, 1 … Σ 43, 364, 3541, 38642, 465599, 6128472, …
  (−∞, 0]nonnegative
• A137597 → x ↦ x+1 → x ↦ x+1
  1 · 4, 1 · 19, 9, 1 · 103, 70, 15, 1 … Σ 1, 5, 29, 189, 1357, 10589, 88909, …
  (−∞, −1]nonnegative
• basis-map coefficients of A297191, shifted → (θ+1)² → θ+1
  1 · 0, 0, 27 · 0, 0, 0, 0, 125 … Σ 1, 27, 125, 343, 729, 1331, 2197, 3375, …
  (−∞, 0]nonnegativerational roots
• A124324 → D
  1 · 4 · 11, 6 · 26, 50 … Σ 0, 0, 1, 4, 17, 76, 362, 1842, …
  (−∞, 0]nonnegative
• A001263 → D → θ+1
  1 · 3, 4 · 6, 24, 9 · 10, 80, 90, 16 … Σ 0, 1, 7, 39, 196, 930, 4257, 19019, …
  (−∞, 0]nonnegative
• A028338 → basis A008277 → θ+1
  1 · 2, 2 · 8, 14, 3 · 48, 114, 45, 4 … Σ 1, 4, 25, 211, 2236, 28471, 422899, …
  (−∞, 0]nonnegative
• A266326: Triangle read by rows giving successive states of cellular automaton generated by "Rule 20" initiated with a single ON (black) cell.
  A133081 → reverse (shift +1)
  1 · 0, 1, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 1, 1 … Σ 1, 2, 1, 2, 1, 2, 1, 2, …
  (−∞, 0]nonnegativerational roots
• A064861 → x ↦ x+1
  6, 7, 2 · 18, 33, 20, 4 · 36, 84, 73, 28, 4 … Σ 15, 75, 225, 1125, 3375, 16875, 50625, …
  (−∞, −1]nonnegativerational roots
• A059364 → basis A013609 → θ+1
  1 · 3, 4 · 15, 44, 24 · 105, 488, 564, 192 … Σ 1, 7, 83, 1349, 27613, 679323, …
  (−∞, 0]nonnegative
• A013612 → x ↦ x+1 → x ↦ x/(x+1)
  36, 132, 121 · 216, 1188, 2178, 1331 … Σ 289, 4913, 83521, 1419857, 24137569, …
  [−1, 0]nonnegativerational roots
• A100862 → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 18, 9, 1 … Σ 1, 3, 10, 35, 129, 497, 1996, 8325, …
  (−∞, 0]nonnegative
• A327997 → D
  1 · 7, 2 · 38, 24, 3 · 192, 202, 54, 4 … Σ 0, 1, 9, 65, 452, 3229, 24489, 200745, …
  (−∞, 0]nonnegative
• A081578 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 7, 28, 25 · 20, 120, 228, 136 … Σ 1, 6, 60, 504, 4464, 38880, 340416, …
  [−1, 0]nonnegative
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
  1 · -1, -2 · 1, 4, 4 · -1, -6, -12, -8 … Σ 1, -3, 9, -27, 81, -243, 729, -2187, …
  [−1, 0]rational roots
• A081582 → γ → θ+1
  1 · 1 · 1, 14 · 1, 28 … Σ 1, 1, 15, 29, 190, 498, 2325, 7043, …
  (−∞, 0]nonnegative