Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• basis-map coefficients of A112227 → basis A013610
  1, 6, 9 · 1, 9, 27, 27 · 1, 12, 54, 108, 81 … Σ 16, 64, 256, 1024, 4096, 16384, 65536, …
  [−1, 0]nonnegativerational roots
• A178821 → basis A013609 → (θ+1)²
  1 · 10, 40 · 60, 480, 540 · 280, 3360, 7560, 4480 … Σ 1, 50, 1080, 15680, 179200, 1741824, …
  (−∞, 0]nonnegative
• A034870 → x ↦ x+1 → θ+1
  1 · 4, 8, 3 · 16, 64, 72, 32, 5 … Σ 1, 15, 189, 2187, 24057, 255879, …
  (−∞, 0]nonnegativerational roots
• A290598 → reverse
  1 · 1, 4 · 1, 14, 28 · 1, 30, 210, 280 … Σ 1, 5, 43, 521, 8113, 154001, 3442915, …
  (−∞, 0]nonnegative
• A038319 → basis A013609 → reverse
  100, 320, 256 · 1000, 4800, 7680, 4096 … Σ 676, 17576, 456976, 11881376, …
  (−∞, 0]nonnegativerational roots
• A091044 → D → D+1
  2 · 16, 6 · 84, 80, 12 · 312, 612, 240, 20 … Σ 0, 2, 22, 176, 1184, 7168, 40448, …
  (−∞, 0]nonnegative
• A331333 → D+1
  1 · 3, 2 · 10, 16, 4 · 42, 108, 60, 8 … Σ 1, 5, 30, 218, 1864, 18312, 202960, …
  (−∞, 0]nonnegative
• A207607 → basis A007318 → θ+1
  1 · 3, 4 · 8, 18, 6 · 21, 66, 45, 8 … Σ 1, 7, 32, 140, 599, 2521, 10474, 43066, …
  (−∞, 0]nonnegative
• A173018 → reverse (shift +1) → θ+1 → reverse
  1 · 2 · 3, 2 · 4, 12, 2 … Σ 1, 2, 5, 18, 84, 480, 3240, 25200, …
  (−∞, 0]nonnegative
• A136572 → basis A013610 → (θ+1)²
  1 · 1, 12 · 2, 48, 162 · 6, 216, 1458, 2592 … Σ 1, 13, 212, 4272, 102912, 2887680, …
  (−∞, 0]nonnegative
• A128100 → basis A013610 → D
  3 · 6 · 21, 18 · 48, 54 … Σ 0, 0, 3, 6, 39, 102, 420, 1218, …
  (−∞, 0]nonnegative
• A038325 → basis A013609 → D
  22 · 968, 968 · 31944, 63888, 31944 … Σ 0, 22, 1936, 127776, 7496192, …
  (−∞, 0]nonnegativerational roots
• A178822 → Veronese 2 (j=1) → x ↦ x+1
  6 · 42 · 224, 56 · 1008, 504 … Σ 0, 6, 42, 280, 1512, 7308, 32340, …
  (−∞, −1]nonnegative
• A059344 → basis A013610 → θ+1
  1 · 1 · 3, 12 · 7, 36 … Σ 1, 1, 15, 43, 565, 2541, 32371, 191815, …
  (−∞, 0]nonnegative
• A038226 → basis A007318 → x ↦ x/(x+1)
  121, 418, 361 · 1331, 6897, 11913, 6859 … Σ 900, 27000, 810000, 24300000, …
  [−1, 0]nonnegativerational roots
• A134346 → D+1
  1 · 6, 3 · 21, 28, 7 · 60, 135, 90, 15 … Σ 1, 9, 56, 300, 1488, 7056, 32512, …
  (−∞, 0]nonnegativerational roots
• A038243 → D+1
  1 · 6, 1 · 35, 12, 1 · 200, 105, 18, 1 … Σ 1, 7, 48, 324, 2160, 14256, 93312, …
  (−∞, 0]nonnegativerational roots
• A094638 → basis A013610 → reverse
  1 · 3, 2 · 18, 21, 6 · 162, 261, 138, 24 … Σ 1, 5, 45, 585, 9945, 208845, 5221125, …
  (−∞, 0]nonnegativerational roots
• A013613 → D+1
  1 · 7, 6 · 13, 84, 36 · 19, 234, 756, 216 … Σ 1, 13, 133, 1225, 10633, 88837, 722701, …
  (−∞, 0]nonnegativerational roots
• A200536 → basis A007318 → x ↦ x/(x+1)
  1 · 6, 19, 15 · 36, 228, 541, 570, 225 … Σ 1, 40, 1600, 64000, 2560000, 102400000, …
  [−1, 0]nonnegativerational roots
• A395972 → (θ+1)²
  4, 64, 72 · 8, 288, 1296, 768 … Σ 140, 2360, 45584, 986272, 23577280, …
  (−∞, 0]nonnegative
• A219512 → x ↦ x+1
  1 · 4, 3 · 28, 42, 15 · 280, 630, 456, 105 … Σ 1, 7, 85, 1471, 33061, 914047, …
  (−∞, −1]nonnegative
• A207823 → basis A013610 → (θ+1)²
  1 · 5, 12 · 24, 120, 81 · 115, 876, 1215, 432 … Σ 1, 17, 225, 2638, 28763, 298611, …
  (−∞, 0]nonnegative
• A127648 → (θ+1)²
  1 · 0, 8 · 0, 0, 27 · 0, 0, 0, 64 … Σ 0, 1, 8, 27, 64, 125, 216, 343, …
  (−∞, 0]nonnegativerational roots
• A063967 → basis A007318 → D
  1 · 5, 2 · 20, 16, 3 · 72, 86, 33, 4 … Σ 0, 1, 7, 39, 195, 918, 4158, 18333, …
  (−∞, 0]nonnegative
• A091042 → basis A007318 → reverse
  1 · 3, 4 · 5, 20, 16 · 7, 56, 112, 64 … Σ 1, 7, 41, 239, 1393, 8119, 47321, …
  (−∞, 0]nonnegative
• A161198 → D
  2 · 8, 8 · 46, 72, 24 · 352, 688, 384, 64 … Σ 0, 2, 16, 142, 1488, 18258, 258144, …
  (−∞, 0]nonnegative
• A036969 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 24, 59, 37 … Σ 1, 3, 15, 121, 1345, 19371, 351663, …
  [−1, 0]nonnegative
• A003506 → Veronese 2 (j=1) → x ↦ x+1
  2 · 6 · 16, 4 · 40, 20 … Σ 0, 0, 2, 6, 20, 60, 174, 490, …
  (−∞, −1]nonnegative
• A257606 → D+1
  1 · 8, 4 · 56, 72, 16 · 360, 888, 488, 64 … Σ 1, 12, 144, 1800, 23760, 332640, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, 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
• A142461 → x ↦ x+1
  1 · 2, 1 · 16, 16, 1 · 224, 336, 114, 1 … Σ 1, 3, 33, 675, 19521, 732483, 33821793, …
  (−∞, −1]nonnegative
• A013622 → D+1
  1 · 8, 5 · 39, 80, 25 · 162, 585, 600, 125 … Σ 1, 13, 144, 1472, 14336, 135168, …
  (−∞, 0]nonnegativerational roots
• A001263 → basis A013611 → (θ+1)²
  1 · 2, 16 · 5, 80, 144 · 14, 336, 1296, 1024 … Σ 1, 18, 229, 2670, 30186, 336356, …
  (−∞, 0]nonnegative
• A154986 → D+1 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 16, 12 · 12, 69, 116, 60 … Σ 1, 5, 33, 257, 2337, 24365, 286977, …
  [−1, 0]nonnegative
• A132774 → θ+1
  1 · 2, 6 · 0, 8, 15 · 0, 0, 18, 28 … Σ 1, 8, 23, 46, 77, 116, 163, 218, …
  (−∞, 0]nonnegativerational roots
• A132393 → basis A008277 → x ↦ x+1
  1 · 2, 1 · 7, 6, 1 · 34, 39, 12, 1 … Σ 1, 3, 14, 86, 648, 5752, 58576, 671568, …
  (−∞, −1]nonnegative
• A249139 → basis A013610 → D+1
  1 · 7, 3 · 13, 6 · 46, 45, 9 … Σ 1, 10, 19, 100, 253, 1018, 2935, 10432, …
  (−∞, 0]nonnegative
• A126216 → D+1
  1 · 3, 1 · 10, 7, 1 · 35, 39, 12, 1 … Σ 1, 4, 18, 87, 439, 2278, 12052, 64669, …
  (−∞, 0]nonnegative
• A132774 → Veronese 2 → D
  5 · 6 · 0, 18 · 0, 20 … Σ 0, 0, 5, 6, 18, 20, 39, 42, …
  (−∞, 0]nonnegativerational roots
• A013620 → basis A013611 → D+1
  1 · 17, 12 · 145, 408, 144 · 1025, 5220, 7344, 1728 … Σ 1, 29, 697, 15317, 319345, 6431117, …
  (−∞, 0]nonnegativerational roots
• A121574 → x ↦ x+1
  1 · 3, 1 · 10, 7, 1 · 33, 35, 11, 1 … Σ 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  (−∞, −1]nonnegative
• A019538 → θ+1
  1 · 1, 4 · 1, 12, 18 · 1, 28, 108, 96 … Σ 1, 5, 31, 233, 2071, 21305, 249271, …
  (−∞, 0]nonnegative
• A257627 → x ↦ x+1
  1 · 6, 3 · 78, 78, 9 · 1560, 2340, 834, 27 … Σ 1, 9, 165, 4761, 185565, 9103329, …
  (−∞, −1]nonnegative
• A147716 → basis A013610 → reverse
  1 · 3, 15 · 9, 90, 225 · 27, 405, 2025, 3375 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, 0]nonnegativerational roots
• A038330 → reverse
  16, 96, 144 · 64, 576, 1728, 1728 … Σ 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0]nonnegativerational roots
• A152842 → reverse → D
  1 · 4, 2 · 15, 14, 3 · 24, 44, 24, 4 … Σ 0, 1, 6, 32, 96, 448, 1152, 5120, …
  (−∞, 0]nonnegativerational roots
• A122848 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 0, 9, 4 … Σ 1, 2, 5, 13, 38, 116, 382, 1310, …
  (−∞, 0]nonnegative
• A113953: A Jacobsthal triangle.
  1 · 0, 1 · 0, 2, 1 · 0, 0, 4, 1 … Σ 1, 1, 3, 5, 11, 21, 43, 85, …
  (−8, 0]nonnegative
• A038331 → basis A013611 → basis A007318
  1369, 1480, 400 · 50653, 82140, 44400, 8000 … Σ 3249, 185193, 10556001, 601692057, …
  (−∞, −1]nonnegativerational roots