Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A135065: A127733 * A007318 as infinite lower triangular matrices.
  1 · 4, 4 · 9, 18, 9 · 16, 48, 48, 16 … Σ 1, 8, 36, 128, 400, 1152, 3136, 8192, …
  (−∞, −1]nonnegativerational roots
• A257621 → D+1
  1 · 6, 3 · 51, 60, 9 · 420, 1179, 474, 27 … Σ 1, 9, 120, 2100, 45360, 1164240, …
  (−∞, 0]nonnegative
• A257612 → x ↦ x+1
  1 · 4, 2 · 32, 32, 4 · 384, 576, 208, 8 … Σ 1, 6, 68, 1176, 27152, 783456, …
  (−∞, −1]nonnegative
• A027467 → basis A013609 → D+1
  1 · 18, 2 · 320, 72, 4 · 5632, 1920, 216, 8 … Σ 1, 20, 396, 7776, 151632, 2939328, …
  (−∞, 0]nonnegativerational roots
• A028326 → basis A013610 → x ↦ x/(x+1)
  2 · 4, 10 · 8, 40, 50 · 16, 120, 300, 250 … Σ 2, 14, 98, 686, 4802, 33614, 235298, …
  [−1, 0]nonnegativerational roots
• A130534 → D²∘x (shift +1) → θ+1
  1 · 3, 6 · 11, 36, 18 · 50, 210, 180, 40 … Σ 1, 9, 65, 480, 3829, 33439, 319976, …
  (−∞, 0]nonnegative
• A038259 → basis A013611 → D+1
  1 · 31, 20 · 561, 1240, 400 … Σ 1, 51, 2201, 87451, 3306801, 120981251, …
  (−∞, 0]nonnegativerational roots
• A038321 → basis A013609 → D+1
  828, 896, 196 · 19440, 34776, 18816, 2744 … Σ 1920, 75776, 2883584, 106954752, …
  (−∞, 0]nonnegativerational roots
• A008292 → γ (scale 1/4) (shift +1) → θ+1
  1 · 2 · 4, 4 · 8, 32 … Σ 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0]nonnegative
• A094816 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 19, 18, 4 · 106, 142, 60, 8 … Σ 1, 6, 41, 316, 2721, 25946, 271801, …
  (−∞, −1]nonnegative
• A269945 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 2, 15, 4 … Σ 1, 2, 5, 21, 126, 1001, 10129, 126006, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013609 → θ+1
  1 · 1, 8, 12 · 1, 16, 72, 128, 80 … Σ 1, 21, 297, 3645, 41553, 452709, …
  (−∞, 0]nonnegativerational roots
• A249248 → reverse → θ+1
  1 · 3, 2 · 3, 14, 3 · 3, 44, 36, 4 … Σ 1, 5, 20, 87, 367, 1615, 7206, 33149, …
  (−∞, 0]nonnegative
• A132393 → basis A013611 → x ↦ x+1
  1 · 5, 4 · 30, 44, 16 · 210, 428, 288, 64 … Σ 1, 9, 90, 990, 11880, 154440, 2162160, …
  (−∞, −1]nonnegativerational roots
• A130534 → basis A008277 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 24, 108, 156, 73 … Σ 1, 5, 37, 361, 4361, 62701, 1044205, …
  [−1, 0]nonnegative
• A135065 → x ↦ x+1 → D+1
  1 · 12, 4 · 72, 54, 9 · 320, 384, 144, 16 … Σ 1, 16, 135, 864, 4725, 23328, 107163, …
  (−∞, 0]nonnegativerational roots
• A038320 → x ↦ x/(x+1)
  121, 374, 289 · 1331, 6171, 9537, 4913 … Σ 784, 21952, 614656, 17210368, …
  [−1, 0]nonnegativerational roots
• A038303 → basis A013610 → reverse
  1 · 3, 11 · 9, 66, 121 · 27, 297, 1089, 1331 … Σ 1, 14, 196, 2744, 38416, 537824, …
  (−∞, 0]nonnegativerational roots
• A013620 → Veronese 2 → D+1
  1 · 2 · 13, 9 · 62, 54 … Σ 1, 2, 22, 116, 691, 3902, 21880, …
  (−∞, 0]nonnegative
• A063967 → x ↦ x−12
  1 · -11, 1 · 110, -21, 1 · -1089, 319, -31, 1 … Σ 1, -10, 90, -800, 7100, -63000, 559000, …
  (7, 11]
• A201701 → D+1
  1 · 1 · 3, 1 · 7, 3 … Σ 1, 1, 4, 10, 27, 71, 186, 484, …
  (−∞, 0]nonnegative
• A173018 → reverse (shift +1) → reverse → θ+1
  1 · 1 · 1, 2 · 1, 8, 3 … Σ 1, 1, 3, 12, 60, 360, 2520, 20160, …
  (−∞, 0]nonnegative
• A193685 → D+1
  1 · 6, 1 · 36, 13, 1 · 216, 127, 21, 1 … Σ 1, 7, 50, 365, 2727, 20878, 163967, …
  (−∞, 0]nonnegative
• A143495 → D+1 → θ+1
  1 · 4, 2 · 16, 18, 3 · 64, 122, 45, 4 … Σ 1, 6, 37, 235, 1540, 10427, 73013, …
  (−∞, 0]nonnegative
• A137320 → x ↦ x+1 → θ+1
  1 · 2, 4 · 6, 20, 12 · 24, 104, 108, 32 … Σ 1, 6, 38, 268, 2116, 18608, 180912, …
  (−∞, 0]nonnegative
• A344565 → (θ+1)²
  3 · 15, 60 · 45, 360, 405 · 105, 1260, 2835, 1680 … Σ 3, 75, 810, 5880, 33600, 163296, …
  (−∞, 0]nonnegative
• A104684 → D → (θ+1)²
  1 · 6, 8 · 30, 96, 27 · 140, 720, 540, 64 … Σ 0, 1, 14, 153, 1464, 12825, 105678, …
  (−∞, 0]nonnegative
• A046716 → basis A013610 → reverse
  1 · 3, 2 · 9, 15, 5 · 27, 99, 75, 16 … Σ 1, 5, 29, 217, 2297, 34349, 674965, …
  (−∞, 0]nonnegative
• A288874: Row reversed version of triangle A201637 (second-order Eulerian triangle).
  A201637 → reverse (shift +1)
  1 · 0, 1 · 0, 2, 1 · 0, 6, 8, 1 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0]nonnegative
• A128813 → basis A007318 → D+1
  1 · 3, 1 · 18, 16, 3 · 174, 280, 135, 18 … Σ 1, 4, 37, 607, 15477, 565782, 28060809, …
  (−∞, 0]nonnegative
• A038320 → D+1
  253, 204, 36 · 3509, 4554, 1836, 216 … Σ 493, 10115, 201433, 3925487, 75252421, …
  (−∞, 0]nonnegativerational roots
• A094816 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 29, 33, 9 · 193, 312, 162, 27 … Σ 1, 8, 71, 694, 7421, 86276, 1084483, …
  (−∞, −1]nonnegative
• A104684 → basis A013609 → θ+1
  1 · 3, 4 · 13, 32, 12 · 63, 228, 180, 32 … Σ 1, 7, 57, 503, 4577, 42135, 389625, …
  (−∞, 0]nonnegative
• A085881 → γ
  1 · 1 · 3 · 15 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+2
• A038323 → D
  9 · 198, 162 · 3267, 5346, 2187 … Σ 0, 9, 360, 10800, 288000, 7200000, …
  (−∞, 0]nonnegativerational roots
• A113953 → basis A013611 → θ+1
  1 · 1, 8 · 3, 32, 48 · 5, 88, 336, 256 … Σ 1, 9, 83, 685, 5435, 41725, 313275, …
  (−∞, 0]nonnegative
• A114656 → basis A013609 → D+1
  1 · 5, 2 · 27, 24, 4 · 147, 222, 84, 8 … Σ 1, 7, 55, 461, 4013, 35795, 324715, …
  (−∞, 0]nonnegative
• A229032 → x ↦ x+1 → θ+1
  1 · 2 · 7, 8 · 20, 32 … Σ 1, 2, 15, 52, 253, 1014, 4395, 18088, …
  (−∞, 0]nonnegative
• A391403 → x ↦ x+1 → D+1
  1 · 1 · 5, 2 · 13, 6 … Σ 1, 1, 7, 19, 85, 301, 1211, 4607, …
  (−∞, 0]nonnegative
• A038220 → basis A013610 → basis A013610
  1 · 11, 18 · 121, 396, 324 · 1331, 6534, 10692, 5832 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  (−∞, 0]nonnegativerational roots
• A136124 → x ↦ x+1 → D
  1 · 7, 2 · 47, 24, 3 · 342, 238, 54, 4 … Σ 0, 1, 9, 74, 638, 5944, 60216, 662640, …
  (−∞, 0]nonnegative
• A075497 → D+1 → D+1
  1 · 4, 1 · 18, 10, 1 · 88, 82, 18, 1 … Σ 1, 5, 29, 189, 1357, 10601, 89329, …
  (−∞, 0]nonnegative
• A038221 → basis A013609 → θ+1
  1 · 6, 12 · 36, 144, 108 · 216, 1296, 1944, 864 … Σ 1, 18, 288, 4320, 62208, 870912, …
  (−∞, 0]nonnegativerational roots
• A136572 → (θ+1)² → D
  4 · 0, 36 · 0, 0, 288 · 0, 0, 0, 2400 … Σ 0, 4, 36, 288, 2400, 21600, 211680, …
  (−∞, 0]nonnegativerational roots
• A038279 → basis A013611 → x ↦ x/(x+1)
  1 · 9, 13 · 81, 234, 169 · 729, 3159, 4563, 2197 … Σ 1, 22, 484, 10648, 234256, 5153632, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A026729, shifted → (θ+1)²
  1 · 0, -4 · 0, 0, 9 · 0, 0, 0, -16 … Σ 1, -4, 9, -16, 25, -36, 49, -64, …
  [−1, 0]rational roots
• A053117 → x ↦ x+1 → D
  2 · 8, 8 · 20, 48, 24 · 40, 168, 192, 64 … Σ 0, 2, 16, 92, 464, 2182, 9824, 42936, …
  (−∞, 0]nonnegative
• A038305 → basis A013610 → (θ+1)²
  169, 936, 729 · 2197, 18252, 28431, 11664 … Σ 1834, 60544, 1854688, 54027952, …
  (−∞, 0]nonnegative
• A154986 → D+1 → x ↦ x+1
  1 · 3, 1 · 12, 8, 1 · 60, 64, 17, 1 … Σ 1, 4, 21, 142, 1161, 11200, 123981, …
  (−∞, −1]nonnegative
• A013620 → basis A013610 → reverse
  1 · 9, 5 · 81, 90, 25 · 729, 1215, 675, 125 … Σ 1, 14, 196, 2744, 38416, 537824, …
  (−∞, 0]nonnegativerational roots