Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A091042 → (θ+1)²
  1 · 1, 12 · 1, 40, 45 · 1, 84, 315, 112 … Σ 1, 13, 86, 512, 2848, 15104, 77312, …
  (−∞, 0]nonnegative
• A026374 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 8, 12, 4 … Σ 1, 3, 10, 25, 75, 175, 500, 1125, …
  (−∞, 0]nonnegative
• A038271 → basis A007318 → basis A007318
  1 · 17, 5 · 289, 170, 25 · 4913, 4335, 1275, 125 … Σ 1, 22, 484, 10648, 234256, 5153632, …
  (−∞, −1]nonnegativerational roots
• A038333 → D+1
  312, 266, 49 · 4752, 6552, 2793, 343 … Σ 627, 14440, 322373, 7037334, 151042039, …
  (−∞, 0]nonnegativerational roots
• A290316 → (θ+1)²
  1 · 1, 24 · 1, 192, 810 · 1, 936, 19926, 34560 … Σ 1, 25, 1003, 55423, 3866281, 324072841, …
  (−∞, 0]nonnegative
• A038226 → basis A007318 → (θ+1)²
  121, 704, 576 · 1331, 11616, 19008, 8192 … Σ 1401, 40147, 1066033, 26894139, …
  (−∞, 0]nonnegative
• A137320 → reverse → (θ+1)²
  1 · 2 · 4, 8 · 8, 48, 36 … Σ 1, 2, 12, 92, 796, 7592, 79064, 893440, …
  (−∞, 0]nonnegative
• A038208 → basis A013610 → (θ+1)²
  1 · 4, 24 · 16, 192, 324 · 64, 1152, 3888, 3456 … Σ 1, 28, 532, 8560, 125200, 1720000, …
  (−∞, 0]nonnegative
• A099097 → basis A013609 → basis A007318
  1 · 9, 6 · 84, 110, 36 · 783, 1530, 996, 216 … Σ 1, 15, 230, 3525, 54025, 828000, …
  (−∞, −1]nonnegative
• A143491 → basis A013610 → θ+1
  12, 42, 27 · 60, 282, 324, 108 … Σ 81, 774, 7974, 88812, 1067652, …
  (−∞, 0]nonnegative
• A152198 → basis A013610 → D+1
  1 · 1 · 5, 3 · 5, 3 … Σ 1, 1, 8, 8, 55, 55, 350, 350, …
  (−∞, 0]nonnegativerational roots
• A038255 → basis A013611 → D+1
  1 · 11, 4 · 105, 88, 16 · 931, 1260, 528, 64 … Σ 1, 15, 209, 2783, 35937, 453871, …
  (−∞, 0]nonnegativerational roots
• A063967 → basis A013610 → θ+1
  1 · 2, 6 · 6, 30, 27 · 16, 120, 216, 108 … Σ 1, 8, 63, 460, 3230, 22050, 147475, …
  (−∞, 0]nonnegative
• A081578 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 7, 7 · 1, 12, 30, 20 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, 0]nonnegative
• A078812 → basis A013609 → (θ+1)²
  1 · 3, 8 · 8, 48, 36 · 21, 200, 324, 128 … Σ 1, 11, 92, 673, 4547, 29128, 179557, …
  (−∞, 0]nonnegative
• A093375 → (θ+1)²
  1 · 2, 4 · 3, 16, 9 · 4, 36, 54, 16 … Σ 1, 6, 28, 110, 384, 1232, 3712, 10656, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → D
  -1 · 0, 4 · 0, 0, -18 · 0, 0, 0, 96 … Σ 0, -1, 4, -18, 96, -600, 4320, -35280, …
  (−∞, 0]rational roots
• A225473 → x ↦ x+1
  1 · 7, 4 · 81, 104, 32 · 1399, 2812, 1824, 384 … Σ 1, 11, 217, 6419, 253297, 12494171, …
  (−2, −1)nonnegative
• A159854 → basis A013611 → (θ+1)²
  1, 32, 144 · 2, 80, 576, 1024 … Σ 177, 1682, 14612, 119592, 937872, …
  (−∞, 0]nonnegative
• A274498 → x ↦ x+1 → D+1
  1 · 3 · 15, 6 · 45, 18 … Σ 1, 3, 21, 63, 405, 1215, 7425, 22275, …
  (−∞, 0]nonnegativerational roots
• A099097 → basis A007318 → basis A007318
  1 · 6, 3 · 38, 37, 9 · 240, 348, 168, 27 … Σ 1, 9, 84, 783, 7299, 68040, 634257, …
  (−∞, −1]nonnegative
• A156319 → (θ+1)²
  1 · 2, 4 · 0, 8, 9 · 0, 0, 18, 16 … Σ 1, 6, 17, 34, 57, 86, 121, 162, …
  (−∞, 0]nonnegativerational roots
• A123160 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 8 · 1, 12, 39, 38 … Σ 1, 3, 15, 90, 579, 3858, 26262, 181380, …
  [−1, 0]nonnegative
• A038321 → reverse
  49, 154, 121 · 343, 1617, 2541, 1331 … Σ 324, 5832, 104976, 1889568, 34012224, …
  (−∞, 0]nonnegativerational roots
• A308737 → D
  3 · 8, 14 · 17, 62, 45 · 34, 192, 282, 124 … Σ 0, 3, 22, 124, 632, 3056, 14304, 65472, …
  (−∞, 0]nonnegativerational roots
• A013609 → D → θ+1
  2 · 4, 16 · 6, 48, 72 · 8, 96, 288, 256 … Σ 0, 2, 20, 126, 648, 2970, 12636, 51030, …
  (−∞, 0]nonnegativerational roots
• A084950 → basis A007318
  1 · 3, 1 · 10, 4 · 43, 20, 1 … Σ 1, 4, 14, 64, 348, 2216, 16208, 134096, …
  (−∞, −1]nonnegative
• A038320 → basis A013609 → x ↦ x/(x+1)
  289, 986, 841 · 4913, 25143, 42891, 24389 … Σ 2116, 97336, 4477456, 205962976, …
  [−1, 0]nonnegativerational roots
• A038328 → x ↦ x+1 → D+1
  252, 64, 4 · 3920, 1512, 192, 8 … Σ 320, 5632, 98304, 1703936, 29360128, …
  (−∞, 0]nonnegativerational roots
• A257607 → x ↦ x+1
  1 · 10, 5 · 110, 110, 25 · 1320, 1980, 910, 125 … Σ 1, 15, 245, 4335, 82685, 1692375, …
  (−∞, −1]nonnegative
• A113413 → basis A013609 → D
  2 · 12, 8 · 46, 72, 24 · 152, 384, 288, 64 … Σ 0, 2, 20, 142, 888, 5186, 29020, …
  (−∞, 0]nonnegative
• A133214 → basis A007318 → reverse
  4, 16, 13 · 8, 60, 114, 63 · 16, 192, 624, 768, 321 … Σ 33, 245, 1921, 15525, 127905, 1067925, …
  (−∞, 0]nonnegative
• A257613 → D+1
  1 · 8, 4 · 64, 80, 16 · 480, 1248, 608, 64 … Σ 1, 12, 160, 2400, 40320, 752640, …
  (−∞, 0]nonnegative
• A093375 → Veronese 2 (j=1) → D
  1 · 8 · 30, 2 · 80, 24 … Σ 0, 0, 0, 1, 8, 32, 104, 304, …
  (−∞, 0]nonnegative
• Veronese parent of A109446 → θ+1
  0, 2 · 1, 2 · 0, 2, 6, 4 · 1, 6, 9, 4 … Σ 2, 3, 12, 20, 64, 112, 320, 576, …
  (−∞, 0]nonnegativerational roots
• A178821 → Veronese 2 → x ↦ x+1
  1 · 5 · 30, 15 · 140, 105 … Σ 1, 5, 45, 245, 1190, 5166, 20790, …
  (−∞, −1]nonnegative
• A038301 → reverse → (θ+1)²
  121, 792, 729 · 1331, 13068, 24057, 11664 … Σ 1642, 50120, 1412800, 37760000, …
  (−∞, 0]nonnegative
• A225470 → x ↦ x/(x+1)
  1 · 2, 3 · 10, 27, 18 · 80, 306, 387, 162 … Σ 1, 5, 55, 935, 21505, 623645, 21827575, …
  [−1, 0]nonnegativerational roots
• A123519 → reverse → D
  6, 2 · 20, 24, 3 · 56, 120, 60, 4 … Σ 8, 47, 240, 1137, 5144, 22559, 96736, …
  (−∞, 0]nonnegative
• A130123 → (θ+1)² → D
  8 · 0, 72 · 0, 0, 384 · 0, 0, 0, 1600 … Σ 0, 8, 72, 384, 1600, 5760, 18816, …
  (−∞, 0]nonnegativerational roots
• A178821 → Veronese 2 → D
  15 · 105 · 420, 140 · 1260, 1260 … Σ 0, 0, 15, 105, 560, 2520, 10080, 36960, …
  (−∞, 0]nonnegative
• A081582 → D
  1 · 9, 2 · 17, 34, 3 · 25, 194, 75, 4 … Σ 0, 1, 11, 54, 298, 1375, 6429, 28476, …
  (−∞, 0]nonnegative
• A063967 → basis A008277
  1 · 2, 1 · 6, 6, 1 · 16, 29, 11, 1 … Σ 1, 3, 13, 57, 272, 1389, 7576, 43929, …
  (−∞, 0]nonnegative
• A046716 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 16, 73, 109, 53 … Σ 1, 5, 31, 251, 2677, 36473, 608995, …
  [−1, 0]nonnegative
• A038226 → D
  48, 128 · 216, 1152, 1536 · 864, 6912, 18432, 16384 … Σ 176, 2904, 42592, 585640, 7730448, …
  (−∞, 0]nonnegativerational roots
• A129710 → D
  1 · 3 · 5, 2 · 7, 8 … Σ 0, 0, 1, 3, 7, 15, 30, 58, …
  (−∞, 0]nonnegative
• A013613 → x ↦ x+1 → reverse
  1 · 6, 7 · 36, 84, 49 · 216, 756, 882, 343 … Σ 1, 13, 169, 2197, 28561, 371293, …
  (−∞, 0]nonnegativerational roots
• A038337 → D
  264, 242 · 4752, 8712, 3993 … Σ 506, 17457, 535348, 15391255, …
  (−∞, 0]nonnegativerational roots
• A038226 → D+1
  57, 176, 64 · 243, 1368, 2112, 512 … Σ 297, 4235, 57233, 746691, 9502009, …
  (−∞, 0]nonnegativerational roots
• A181543: Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 27, 27, 1 … Σ 1, 2, 10, 56, 346, 2252, 15184, 104960, …
  (−∞, 0)nonnegative