Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• basis-map coefficients of A026729 → basis A013611 → D
  4 · 8, 32 · 12, 96, 192 · 16, 192, 768, 1024 … Σ 0, 4, 40, 300, 2000, 12500, 75000, …
  (−∞, 0]nonnegativerational roots
• A322944 → (θ+1)²
  1 · 1, 4 · 2, 24, 9 · 6, 152, 135, 16 … Σ 1, 5, 35, 309, 3277, 40421, 566735, …
  (−∞, 0]nonnegative
• A391533 → Veronese 2 (j=1) → (θ+1)²
  4 · 8, 4 · 12, 24 · 16, 76, 9 … Σ 4, 12, 36, 101, 268, 686, 1708, 4160, …
  (−∞, 0]nonnegative
• A207823 → basis A013611 → reverse
  1 · 4, 5 · 16, 40, 24 · 64, 240, 292, 115 … Σ 1, 9, 80, 711, 6319, 56160, 499121, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → (θ+1)² → (θ+1)²
  0, 0, 81 · 0, 0, 0, 256 · 0, 0, 0, 0, 625 … Σ 81, 256, 625, 1296, 2401, 4096, 6561, …
  (−∞, 0]nonnegativerational roots
• A331333 → θ+1
  1 · 1, 4 · 2, 16, 12 · 6, 72, 108, 32 … Σ 1, 5, 30, 218, 1864, 18312, 202960, …
  (−∞, 0]nonnegative
• A131689 → θ+1
  1 · 0, 2 · 0, 2, 6 · 0, 2, 18, 24 … Σ 1, 2, 8, 44, 308, 2612, 25988, 296564, …
  (−∞, 0]nonnegative
• A038337 → (θ+1)²
  144, 1056, 1089 · 1728, 19008, 39204, 21296 … Σ 2289, 81236, 2653993, 82054248, …
  (−∞, 0]nonnegative
• A128813 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 8, 36, 40 · 56, 404, 964, 760 … Σ 1, 6, 84, 2184, 91728, 5687136, …
  [−1, 0]nonnegativerational roots
• A266213 → basis A013610 → θ+1
  1 · 1 · 3, 12 · 7, 48, 54 … Σ 1, 1, 15, 109, 779, 5321, 35367, …
  (−∞, 0]nonnegative
• A038325 → basis A013609
  1 · 22, 22 · 484, 968, 484 … Σ 1, 44, 1936, 85184, 3748096, 164916224, …
  (−∞, 0]nonnegativerational roots
• A085881 → basis A013611 → D+1
  1 · 6, 4 · 60, 144, 48 · 840, 3600, 4320, 960 … Σ 1, 10, 252, 9720, 498960, 31842720, …
  (−∞, 0]nonnegativerational roots
• A196347 → θ+1
  1 · 1, 2 · 2, 8, 6 · 6, 36, 54, 24 … Σ 1, 3, 16, 120, 1152, 13440, 184320, …
  (−∞, 0]nonnegativerational roots
• A144441 → D
  1 · 14, 2 · 83, 166, 3 · 432, 3100, 1296, 4 … Σ 0, 1, 16, 252, 4832, 110400, 2943552, …
  (−∞, 0]nonnegative
• A225471 → basis A013609
  1 · 4, 2 · 32, 24, 4 · 384, 352, 96, 8 … Σ 1, 6, 60, 840, 15120, 332640, 8648640, …
  (−∞, −1]nonnegativerational roots
• A027465 → Veronese 2 (j=1) → D+1
  1 · 6 · 28, 1 · 120, 12 … Σ 0, 1, 6, 29, 132, 588, 2592, 11344, …
  (−∞, 0]nonnegative
• Veronese parent of A109447 → Veronese 2 (j=1) → x ↦ x+1
  1 · 2 · 4, 3 · 8, 4 … Σ 1, 2, 7, 12, 41, 70, 239, 408, …
  (−∞, −1]nonnegative
• A132393 → θ+1 → reverse
  1 · 2 · 3, 2 · 4, 9, 4 … Σ 1, 2, 5, 17, 74, 394, 2484, 18108, …
  (−∞, 0]nonnegative
• A178822 → basis A013611 → basis A007318
  1 · 36, 24 · 756, 1008, 336 … Σ 1, 60, 2100, 56000, 1260000, 25200000, …
  (−∞, −1]nonnegativerational roots
• A099040 → θ+1 → x ↦ x/(x+1)
  1 · 0, 4 · 0, 4, 16 · 0, 0, 24, 56 … Σ 1, 4, 20, 80, 320, 1216, 4544, 16640, …
  [−1, 0]nonnegative
• A125662 → basis A013611 → D
  4 · 32, 32 · 184, 384, 192 · 928, 2976, 3072, 1024 … Σ 0, 4, 64, 760, 8000, 78860, 745856, …
  (−∞, 0]nonnegative
• A026374 → θ+1 → reverse
  1 · 2, 1 · 3, 6, 1 · 4, 12, 8, 1 … Σ 1, 3, 10, 25, 75, 175, 500, 1125, …
  (−∞, 0]nonnegative
• A078812 → reverse → θ+1
  1 · 1, 4 · 1, 8, 9 · 1, 12, 30, 16 … Σ 1, 5, 18, 59, 185, 564, 1685, 4957, …
  (−∞, 0]nonnegative
• A267482 → D
  1 · 1, 2 · -2, 2, 3 · -2, -6, 3, 4 … Σ 0, 1, 3, 3, -1, -6, -6, 1, …
  (−∞, ∞)
• A133214 → basis A013609 → D
  32, 32 · 228, 480, 192 · 1536, 4992, 4608, 1024 … Σ 64, 900, 12160, 160420, 2081088, …
  (−∞, 0]nonnegative
• A137597 → D+1 → reverse
  1 · 1, 3 · 1, 7, 10 · 1, 12, 40, 37 … Σ 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  (−∞, 0]nonnegative
• A013611 → D+1
  1 · 5, 4 · 9, 40, 16 · 13, 108, 240, 64 … Σ 1, 9, 65, 425, 2625, 15625, 90625, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A112227 → basis A013610 → D
  6, 18 · 9, 54, 81 · 12, 108, 324, 324 … Σ 24, 144, 768, 3840, 18432, 86016, …
  (−∞, 0]nonnegativerational roots
• A120910 → basis A007318 → reverse
  3 · 3, 9 · 3, 18, 27 · 3, 27, 81, 81 … Σ 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, 0]nonnegativerational roots
• A121314 → D
  1 · 1, 2 · 1, 6, 3 · 1, 10, 18, 4 … Σ 0, 1, 3, 10, 33, 105, 324, 977, …
  (−∞, 0]nonnegative
• A166979 → (θ+1)²
  1, 44, 9 · 1, 444, 270, 16 · 1, 4444, 6129, 928, 25 … Σ 54, 731, 11527, 210186, 4355059, …
  (−∞, 0]nonnegative
• A054142 → D → D
  2 · 12, 6 · 30, 60, 12 · 56, 210, 180, 20 … Σ 0, 0, 2, 18, 102, 466, 1884, 7044, …
  (−∞, 0]nonnegative
• A110162 → D
  1 · -4, 2 · 9, -12, 3 · -16, 40, -24, 4 … Σ 0, 1, -2, 0, 4, -5, 0, 7, …
  (−∞, ∞)
• A121314 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 5, 10, 6, 1 … Σ 1, 2, 6, 22, 86, 342, 1366, 5462, …
  (−∞, −1]nonnegative
• basis-map coefficients of A127647 → basis A007318
  1 · 2, 1 · 8, 8, 2 · 24, 36, 18, 3 … Σ 1, 3, 18, 81, 405, 1944, 9477, 45927, …
  (−∞, −1]nonnegativerational roots
• A257625 → (θ+1)²
  1 · 3, 12 · 9, 216, 81 · 27, 2484, 5589, 432 … Σ 1, 15, 306, 8532, 292896, 11897280, …
  (−∞, 0]nonnegative
• A038281 → x ↦ x+1 → D
  66, 18 · 1089, 594, 81 · 15972, 13068, 3564, 324 … Σ 84, 1764, 32928, 576240, 9680832, …
  (−∞, 0]nonnegativerational roots
• A081578 → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 20, 30, 12, 1 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, −1]nonnegative
• A038328 → x ↦ x+1 → (θ+1)²
  196, 224, 36 · 2744, 4704, 1512, 128 … Σ 456, 9088, 176128, 3342336, 62390272, …
  (−∞, 0]nonnegative
• A082985 → reverse → x ↦ x/(x+1)
  1 · 3, 4 · 5, 15, 11 · 7, 35, 56, 29 … Σ 1, 7, 31, 127, 511, 2047, 8191, 32767, …
  [−1, 0)nonnegative
• A143683 → γ → D+1
  1 · 1 · 9, 8 · 17, 16 … Σ 1, 1, 17, 33, 241, 641, 3281, 10209, …
  (−∞, 0]nonnegative
• A359363 → D
  1 · 1, 2 · 1, 8, 3 · 1, 20, 30, 4 … Σ 0, 1, 3, 12, 55, 276, 1477, 8296, …
  (−∞, 0]nonnegative
• A038324 → D+1
  1 · 21, 10 · 341, 420, 100 · 4961, 10230, 6300, 1000 … Σ 1, 31, 861, 22491, 564921, 13808151, …
  (−∞, 0]nonnegativerational roots
• A069466 → basis A007318 → (θ+1)²
  1 · 4, 8 · 36, 144, 54 · 400, 2400, 2160, 320 … Σ 1, 12, 234, 5280, 124950, 3013416, …
  (−∞, 0]nonnegative
• A137320 → Veronese 2 → x ↦ x+1
  1 · 4, 4 · 12, 12 · 60, 76, 16 … Σ 1, 0, 8, 24, 152, 1040, 8144, 71904, …
  (−∞, −1]nonnegative
• A007318 → Veronese 2 (j=1) → x ↦ x+1
  1 · 2 · 4, 1 · 8, 4 … Σ 0, 1, 2, 5, 12, 29, 70, 169, …
  (−∞, −1]nonnegative
• A038301 → x ↦ x+1
  400, 440, 121 · 8000, 13200, 7260, 1331 … Σ 961, 29791, 923521, 28629151, …
  (−∞, −1]nonnegativerational roots
• A085478 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 20, 13 … Σ 1, 3, 11, 43, 171, 683, 2731, 10923, …
  [−1, 0)nonnegative
• A008459 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 18, 27, 4 … Σ 1, 3, 12, 50, 210, 882, 3696, 15444, …
  (−∞, 0]nonnegative
• A286724 → θ+1
  1 · 2, 2 · 8, 16, 3 · 48, 144, 54, 4 … Σ 1, 4, 27, 250, 2917, 40896, 667999, …
  (−∞, 0]nonnegative