Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A113413 → x ↦ x+1 → θ+1
  1 · 3, 2 · 7, 12, 3 · 17, 46, 27, 4 … Σ 1, 5, 22, 94, 390, 1586, 6350, 25114, …
  (−∞, 0]nonnegative
• A106800 → θ+1 → x ↦ x+1
  1 · 1 · 3, 2 · 10, 12, 3 … Σ 1, 1, 5, 25, 141, 901, 6393, 49589, …
  (−∞, −1]nonnegative
• A038279 → basis A013610 → D
  3 · 54, 18 · 729, 486, 81 · 8748, 8748, 2916, 324 … Σ 0, 3, 72, 1296, 20736, 311040, 4478976, …
  (−∞, 0]nonnegativerational roots
• A120910 → basis A013609 → basis A007318
  3 · 15, 6 · 75, 60, 12 · 375, 450, 180, 24 … Σ 3, 21, 147, 1029, 7203, 50421, 352947, …
  (−∞, −1]nonnegativerational roots
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <=…
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … Σ 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, −1]nonnegative
• A124932 → x ↦ x/(x+1)
  1 · 2, 5 · 3, 15, 18 · 4, 30, 72, 56 … Σ 1, 7, 36, 162, 675, 2673, 10206, 37908, …
  [−1, 0]nonnegativerational roots
• A103327 → basis A013610 → D+1
  1 · 7, 3 · 52, 54, 9 · 304, 672, 297, 27 … Σ 1, 10, 115, 1300, 14149, 149470, …
  (−∞, 0]nonnegative
• A008459 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … Σ 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, −1]nonnegative
• A128100 → D
  1 · 2 · 5, 2 · 10, 6 … Σ 0, 0, 1, 2, 7, 16, 41, 94, …
  (−∞, −1]nonnegative
• A114656 → basis A007318 → reverse
  1 · 1, 3 · 1, 8, 11 · 1, 15, 51, 45 … Σ 1, 4, 20, 112, 672, 4224, 27456, …
  (−∞, 0]nonnegative
• A103633 → basis A013609 → D+1
  1 · 3, 2 · 8, 14, 4 · 12, 42, 40, 8 … Σ 1, 5, 26, 102, 480, 1728, 7776, 26784, …
  (−∞, 0]nonnegative
• A038333 → basis A013609 → θ+1
  361, 1064, 588 · 6859, 30324, 33516, 10976 … Σ 2013, 81675, 3198393, 122149863, …
  (−∞, 0]nonnegativerational roots
• A027466 → basis A013610 → (θ+1)²
  1 · 8, 12 · 64, 192, 81 · 512, 2304, 1944, 432 … Σ 1, 20, 337, 5192, 75625, 1059476, …
  (−∞, 0]nonnegative
• A209414 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 28, 81, 16 … Σ 1, 5, 26, 126, 556, 2302, 9108, 34830, …
  (−∞, 0]nonnegative
• A038226 → basis A013611 → D
  704, 2048 · 11616, 67584, 98304 … Σ 2752, 177504, 10176896, 547008160, …
  (−∞, 0]nonnegativerational roots
• A178821 → basis A013611
  1 · 10, 20 · 60, 240, 240 · 280, 1680, 3360, 2240 … Σ 1, 30, 540, 7560, 90720, 979776, …
  [−1, 0]nonnegativerational roots
• A225467 → θ+1
  1 · 3, 8 · 9, 80, 48 · 27, 632, 1008, 256 … Σ 1, 11, 137, 1923, 30097, 519035, …
  (−∞, 0]nonnegative
• A094638 → basis A013611 → θ+1
  1 · 2, 8 · 6, 56, 96 · 24, 368, 1392, 1536 … Σ 1, 10, 158, 3320, 86616, 2695536, …
  (−∞, 0]nonnegative
• A109692 → basis A013610 → reverse
  1 · 3, 2 · 27, 30, 8 · 405, 612, 300, 48 … Σ 1, 5, 65, 1365, 39585, 1464645, …
  (−∞, 0]nonnegativerational roots
• A114656 → basis A013610 → (θ+1)²
  1 · 3, 12 · 11, 96, 81 · 45, 612, 1215, 432 … Σ 1, 15, 188, 2304, 28064, 340656, …
  (−∞, 0]nonnegative
• A144445 → D+1
  1 · 2, 1 · 24, 25, 1 · 207, 618, 209, 1 … Σ 1, 3, 50, 1035, 29091, 1028328, …
  (−∞, 0]nonnegative
• A154602 → D+1
  1 · 2, 1 · 7, 6, 1 · 30, 37, 12, 1 … Σ 1, 3, 14, 80, 536, 4080, 34560, 321088, …
  (−∞, 0]nonnegative
• A144696 → reverse
  1 · 2, 1 · 4, 7, 1 · 8, 33, 18, 1 … Σ 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0]nonnegative
• A120909 → θ+1
  3 · 3, 12 · 3, 24, 36 · 3, 36, 108, 96 … Σ 3, 15, 63, 243, 891, 3159, 10935, …
  (−∞, 0]nonnegativerational roots
• A026374 → D+1
  1 · 2, 1 · 4, 5, 1 · 5, 12, 7, 1 … Σ 1, 3, 10, 25, 75, 175, 500, 1125, …
  (−∞, 0]nonnegative
• A008292 → γ (shift +1) → x ↦ x+1 → D+1
  1 · 1 · 5, 2 · 17, 8 … Σ 1, 1, 7, 25, 195, 1245, 11543, 104657, …
  (−∞, 0]nonnegative
• A156992 → basis A007318 → θ+1
  1 · 4, 4 · 24, 48, 18 · 192, 576, 432, 96 … Σ 1, 8, 90, 1296, 22680, 466560, …
  (−∞, 0]nonnegativerational roots
• A086645 → basis A013611 → reverse
  1 · 4, 2 · 16, 32, 8 · 64, 288, 192, 32 … Σ 1, 6, 56, 576, 6016, 62976, 659456, …
  (−∞, 0]nonnegative
• A085841 → (θ+1)²
  1 · 3, 16 · 5, 160, 144 · 7, 560, 3024, 1024 … Σ 1, 19, 309, 4615, 62761, 795963, …
  (−∞, 0]nonnegative
• A026729 → basis A008277 → x ↦ x+1
  1 · 2, 1 · 7, 6, 1 · 25, 32, 11, 1 … Σ 1, 3, 14, 69, 379, 2264, 14591, 100659, …
  (−∞, −1]nonnegative
• A257614 → θ+1
  1 · 2, 4 · 4, 56, 12 · 8, 488, 732, 32 … Σ 1, 6, 72, 1260, 28728, 804384, …
  (−∞, 0]nonnegative
• A038301 → basis A013611
  400, 1760, 1936 · 8000, 52800, 116160, 85184 … Σ 4096, 262144, 16777216, 1073741824, …
  [−1, 0]nonnegativerational roots
• A114656 → reverse → θ+1
  1 · 1, 4 · 1, 12, 12 · 1, 24, 72, 32 … Σ 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A008288 → θ+1
  1 · 1, 6, 3 · 1, 14, 39, 28, 5 … Σ 1, 10, 87, 676, 4925, 34446, 234227, …
  (−∞, 0]nonnegative
• A142462 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 18, 18 · 1, 146, 432, 288 … Σ 1, 3, 37, 867, 28837, 1248483, …
  [−1, 0]nonnegative
• A038321 → basis A013609 → basis A007318
  1024, 896, 196 · 32768, 43008, 18816, 2744 … Σ 2116, 97336, 4477456, 205962976, …
  (−∞, −1]nonnegativerational roots
• A085881 → basis A013609 → D+1
  1 · 4, 2 · 36, 48, 12 · 480, 1080, 720, 120 … Σ 1, 6, 96, 2400, 80640, 3386880, …
  (−∞, 0]nonnegativerational roots
• A120910 → basis A013609 → reverse
  3 · 6, 9 · 12, 36, 27 · 24, 108, 162, 81 … Σ 3, 15, 75, 375, 1875, 9375, 46875, …
  (−∞, 0]nonnegativerational roots
• A225470 → x ↦ x+1 → reverse
  1 · 1, 3 · 1, 9, 18 · 1, 18, 99, 162 … Σ 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0]nonnegativerational roots
• A132393 → Veronese 2 (j=1) → D
  1 · 6 · 35, 2 · 225, 30 … Σ 0, 0, 0, 1, 6, 37, 255, 1977, …
  (−∞, 0]nonnegative
• A038322 → D+1
  1 · 19, 8 · 297, 304, 64 · 4235, 7128, 3648, 512 … Σ 1, 27, 665, 15523, 349809, 7688939, …
  (−∞, 0]nonnegativerational roots
• A114687 → θ+1 → D
  4 · 12, 24 · 24, 144, 96 · 40, 480, 960, 320 … Σ 0, 4, 36, 264, 1800, 11820, 75852, …
  (−∞, 0]nonnegative
• A038231 → Veronese 2 (j=1) → θ+1
  1 · 8 · 48, 2 · 256, 32 … Σ 0, 1, 8, 50, 288, 1603, 8776, 47604, …
  (−∞, 0]nonnegative
• A038303 → basis A007318 → reverse
  1 · 1, 11 · 1, 22, 121 · 1, 33, 363, 1331 … Σ 1, 12, 144, 1728, 20736, 248832, …
  (−∞, 0]nonnegativerational roots
• A038338 → (θ+1)²
  144, 1152, 1296 · 1728, 20736, 46656, 27648 … Σ 2592, 96768, 3317760, 107495424, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → (θ+1)² → (θ+1)²
  1 · 0, -16 · 0, 0, 81 · 0, 0, 0, -256 … Σ 1, -16, 81, -256, 625, -1296, 2401, …
  (−∞, 0]rational roots
• A038220 → basis A007318 → (θ+1)²
  1 · 5, 8 · 25, 80, 36 · 125, 600, 540, 128 … Σ 1, 13, 141, 1393, 12985, 116277, …
  (−∞, 0]nonnegative
• A113413 → basis A013610 → D
  3 · 18, 18 · 69, 162, 81 · 228, 864, 972, 324 … Σ 0, 3, 36, 312, 2388, 17088, 117228, …
  (−∞, 0]nonnegative
• A386789 → D+1
  1 · 2, 2 · 3, 15, 6 · 4, 44, 80, 20 … Σ 1, 4, 24, 148, 915, 5640, 34622, …
  (−∞, 0]nonnegative
• A038277 → x ↦ x/(x+1) → x ↦ x+1
  1 · 25, 18 · 625, 900, 324 … Σ 1, 43, 1849, 79507, 3418801, 147008443, …
  (−∞, −1]nonnegativerational roots