Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• basis-map coefficients of A026729 → (θ+1)²
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … Σ 1, 4, 9, 16, 25, 36, 49, 64, …
  [−1, 0]nonnegativerational roots
• A052553 → basis A013609 → D
  2 · 4 · 10, 8 · 20, 24 … Σ 0, 0, 2, 4, 18, 44, 136, 348, …
  (−∞, 0]nonnegative
• A034870 → D → (θ+1)²
  2, 8 · 4, 48, 108, 64 · 6, 120, 540, 960, 750, 216 … Σ 0, 10, 224, 2592, 22528, 166400, …
  (−∞, 0]nonnegative
• A038324 → basis A013609 → x ↦ x+1
  1 · 41, 20 · 1681, 1640, 400 … Σ 1, 61, 3721, 226981, 13845841, …
  (−∞, −1]nonnegativerational roots
• A249139 → basis A013610 → x ↦ x+1
  1 · 7, 3 · 13, 6 · 55, 45, 9 … Σ 1, 10, 19, 109, 280, 1261, 3781, 15130, …
  (−∞, −1]nonnegative
• A103633 → Veronese 2 → D
  1 · 1 · 1, 2 · 0, 4 … Σ 0, 0, 1, 1, 3, 4, 9, 11, …
  (−∞, 0]nonnegative
• A048993 → D+1 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 14, 18, 4 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0]nonnegative
• A134346 → θ+1
  1 · 3, 6 · 7, 28, 21 · 15, 90, 135, 60 … Σ 1, 9, 56, 300, 1488, 7056, 32512, …
  (−∞, 0]nonnegativerational roots
• A207606 → basis A013611 → D
  44, 64 · 176, 544, 384 · 620, 2976, 4416, 2048 … Σ 108, 1104, 10060, 86040, 706848, …
  (−∞, 0]nonnegative
• A038326 → x ↦ x+1
  1 · 23, 12 · 529, 552, 144 · 12167, 19044, 9936, 1728 … Σ 1, 35, 1225, 42875, 1500625, 52521875, …
  (−∞, −1]nonnegativerational roots
• A272866 → basis A013610 → x ↦ x/(x+1)
  1 · 5, 25, 29 · 25, 250, 915, 1450, 841 … Σ 1, 59, 3481, 205379, 12117361, …
  [−1, 0]nonnegative
• A225470 → basis A013611 → x ↦ x+1
  1 · 7, 4 · 70, 68, 16 · 910, 1164, 480, 64 … Σ 1, 11, 154, 2618, 52360, 1204280, …
  (−∞, −1]nonnegativerational roots
• A102662 → θ+1
  1 · 1, 6 · 1, 10, 9 · 1, 14, 33, 12 … Σ 1, 7, 20, 60, 171, 477, 1306, 3526, …
  (−∞, 0]nonnegative
• A130123 → basis A013611 → D+1
  1 · 10, 8 · 36, 160, 64 · 104, 864, 1920, 512 … Σ 1, 18, 260, 3400, 42000, 500000, …
  (−∞, 0]nonnegativerational roots
• A173018 → reverse (shift +1) → θ+1 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 5 · 0, 2, 16, 18 … Σ 1, 2, 7, 36, 241, 1978, 19195, 214784, …
  [−1, 0]nonnegative
• A166979 → θ+1
  1, 22, 3 · 1, 222, 90, 4 · 1, 2222, 2043, 232, 5 … Σ 26, 317, 4503, 74074, 1400719, …
  (−∞, 0]nonnegative
• A137320 → reverse → D+1
  1 · 2 · 6, 2 · 20, 20, 4 … Σ 1, 2, 8, 44, 292, 2232, 19224, 184032, …
  (−∞, 0]nonnegative
• A038330 → basis A013611 → x ↦ x/(x+1)
  256, 1024, 1024 · 4096, 24576, 49152, 32768 … Σ 2304, 110592, 5308416, 254803968, …
  [−1, 0]nonnegativerational roots
• image under θ+1 → θ+1 → D+1 → x ↦ x/(x+1)
  1 · 4, 8 · 0, 18, 27 · 0, 0, 48, 64 … Σ 1, 12, 45, 112, 225, 396, 637, 960, …
  [−1, 0]nonnegativerational roots
• A113953 → Veronese 2 → x ↦ x+1
  1 · 1, 1 · 4, 4 · 5, 6, 1 … Σ 1, 0, 2, 8, 12, 32, 104, 224, …
  (−∞, −1]nonnegative
• A207824 → x ↦ x/(x+1)
  1 · 5, 6 · 24, 58, 35 · 115, 418, 506, 204 … Σ 1, 11, 117, 1243, 13205, 140283, …
  [−1, −3/4)nonnegative
• A114656 → θ+1 → D
  2 · 12, 6 · 48, 72, 12 · 160, 480, 240, 20 … Σ 0, 2, 18, 132, 900, 5910, 37926, …
  (−∞, 0]nonnegative
• A147716 → basis A013611 → x ↦ x+1
  1 · 19, 4 · 361, 152, 16 · 6859, 4332, 912, 64 … Σ 1, 23, 529, 12167, 279841, 6436343, …
  (−∞, −1]nonnegativerational roots
• A163626 → D
  -1 · -3, 4 · -7, 24, -18 · -15, 100, -180, 96 … Σ 0, -1, 1, -1, 1, -1, 1, -1, …
  (−∞, ∞)
• basis-map coefficients of A026729 → (θ+1)² → θ+1
  1 · 0, 8 · 0, 0, 27 · 0, 0, 0, 64 … Σ 1, 8, 27, 64, 125, 216, 343, 512, …
  (−∞, 0]nonnegativerational roots
• A114655 → x ↦ x+1 → x ↦ x+1
  2 · 8, 2 · 40, 20, 2 · 224, 168, 36, 2 … Σ 2, 10, 62, 430, 3194, 24850, 199910, …
  (−∞, −1]nonnegative
• basis-map coefficients of A112227 → basis A008277 → (θ+1)²
  1, 12, 9 · 1, 28, 54, 16 · 1, 60, 225, 160, 25 … Σ 22, 99, 471, 2386, 12867, 73681, …
  (−∞, 0]nonnegative
• A063007 → basis A013609 → D
  4 · 36, 48 · 264, 720, 480 · 1800, 7440, 10080, 4480 … Σ 0, 4, 84, 1464, 23800, 373020, 5715948, …
  (−∞, 0]nonnegative
• A229032 → x ↦ x+1 → (θ+1)²
  1 · 2 · 7, 16 · 20, 64 … Σ 1, 2, 23, 84, 493, 2134, 10339, 45928, …
  (−∞, 0]nonnegative
• A132393 → basis A008277
  1 · 1, 1 · 2, 4, 1 · 6, 18, 9, 1 … Σ 1, 2, 7, 34, 209, 1546, 13327, 130922, …
  (−∞, 0]nonnegative
• A201701 → basis A007318
  1 · 1 · 3, 1 · 7, 3 … Σ 1, 1, 4, 10, 28, 76, 208, 568, …
  (−∞, −1]nonnegative
• A207823 → reverse
  1 · 1, 4 · 1, 8, 15 · 1, 12, 46, 56 … Σ 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  (−1/2, 0]nonnegative
• A085881 → basis A007318 → θ+1
  1 · 2, 2 · 12, 24, 9 · 120, 360, 270, 60 … Σ 1, 4, 45, 810, 19845, 612360, 22733865, …
  (−∞, 0]nonnegativerational roots
• A161198 → basis A008277 → D
  2 · 20, 8 · 210, 168, 24 · 2520, 3024, 864, 64 … Σ 0, 2, 28, 402, 6472, 117530, 2389524, …
  (−∞, 0]nonnegative
• A038337 → basis A013609 → D+1
  1541, 1980, 484 · 47081, 101706, 65340, 10648 … Σ 4005, 224775, 12119625, 635596875, …
  (−∞, 0]nonnegativerational roots
• A094816 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 16, 23, 9, 1 … Σ 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  (−∞, −1]nonnegative
• A163626 → x ↦ x+1
  1 · 0, -1 · 0, 1, 2 · 0, -1, -6, -6 … Σ 1, -1, 3, -13, 75, -541, 4683, -47293, …
  (−1, 0]
• A049459: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 3.
  1 · -4, 1 · 20, -9, 1 · -120, 74, -15, 1 … Σ 1, -3, 12, -60, 360, -2520, 20160, …
  [7/2, ∞)rational roots
• A038291 → basis A013611
  1 · 10, 4 · 100, 80, 16 · 1000, 1200, 480, 64 … Σ 1, 14, 196, 2744, 38416, 537824, …
  (−∞, −1]nonnegativerational roots
• A145324 → basis A013610 → (θ+1)²
  1 · 3, 24 · 12, 204, 486 · 60, 1596, 7938, 10368 … Σ 1, 27, 702, 19962, 631917, 22232403, …
  (−∞, 0]nonnegative
• A116412 → basis A013610 → x ↦ x/(x+1)
  1 · 4, 7 · 13, 50, 46 · 43, 255, 489, 304 … Σ 1, 11, 109, 1091, 10909, 109091, …
  [−1, 0]nonnegative
• A145596 → x ↦ x+1 → reverse
  1 · 2, 4 · 3, 14, 14 · 4, 32, 72, 48 … Σ 1, 6, 31, 156, 785, 3978, 20335, …
  (−∞, 0]nonnegative
• A038330 → basis A013610 → x ↦ x/(x+1)
  256, 896, 784 · 4096, 21504, 37632, 21952 … Σ 1936, 85184, 3748096, 164916224, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A026729 → basis A013611 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 25 · 1, 15, 75, 125 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  [−1, 0]nonnegativerational roots
• A126216 → basis A013610 → θ+1
  1 · 3, 6 · 11, 42, 27 · 45, 252, 324, 108 … Σ 1, 9, 80, 729, 6806, 64746, 624694, …
  (−∞, 0]nonnegative
• A281000 → (θ+1)²
  1 · 3, 4 · 10, 40, 9 · 35, 280, 189, 16 … Σ 1, 7, 59, 520, 4675, 42463, 387813, …
  (−∞, 0]nonnegative
• A038323 → basis A013609 → (θ+1)²
  1 · 20, 72 · 400, 2880, 2916 … Σ 1, 92, 6196, 362672, 19551760, …
  (−∞, 0]nonnegative
• A038281 → (θ+1)²
  64, 192, 81 · 512, 2304, 1944, 432 … Σ 337, 5192, 75625, 1059476, 14421385, …
  (−∞, 0]nonnegative
• A178820 → basis A013610 → θ+1
  1 · 8, 24 · 40, 240, 270 · 160, 1440, 3240, 2160 … Σ 1, 32, 550, 7000, 74375, 700000, …
  (−∞, 0]nonnegativerational roots
• A145324 → basis A013610 → D+1
  1 · 9, 6 · 63, 159, 54 · 459, 2163, 2826, 648 … Σ 1, 15, 276, 6096, 157851, 4698117, …
  (−∞, 0]nonnegative