Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A103406 → Veronese 2 (j=1) → D
  4 · 16 · 40, 12 · 80, 72 … Σ 0, 0, 0, 4, 16, 52, 152, 416, …
  (−∞, 0]nonnegative
• A196347 → x ↦ x+1 → θ+1
  1 · 2, 2 · 8, 16, 6 · 48, 144, 108, 24 … Σ 1, 4, 30, 324, 4536, 77760, 1574640, …
  (−∞, 0]nonnegativerational roots
• A111125 → basis A013611 → θ+1
  1 · 4, 8 · 11, 56, 48 · 29, 248, 480, 256 … Σ 1, 12, 115, 1013, 8484, 68711, 543337, …
  (−∞, 0]nonnegative
• A127773 → D → D+1
  3 · 12, 12 · 0, 60, 30 · 0, 0, 180, 60 … Σ 0, 3, 24, 90, 240, 525, 1008, 1764, …
  (−∞, 0]nonnegativerational roots
• A028338: Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  1 · 1, 1 · 3, 4, 1 · 15, 23, 9, 1 … Σ 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  (−∞, 0)nonnegativerational roots
• A034839 → D+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 16, 36, 80, 176, …
  (−∞, 0]nonnegative
• A008292 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 22, 33, 4 … Σ 0, 1, 3, 12, 60, 360, 2520, 20160, …
  (−∞, 0]nonnegative
• A274496 → D → D
  8 · 16 · 48, 48 · 96, 96 … Σ 0, 0, 0, 0, 8, 16, 96, 192, …
  (−∞, 0]nonnegativerational roots
• A105868 → D+1
  1 · 1, 1 · 2, 4, 1 · 0, 12, 9, 1 … Σ 1, 2, 7, 22, 71, 226, 717, 2262, …
  (−∞, 0]nonnegative
• A081580 → basis A007318
  1 · 2, 1 · 9, 9, 1 · 28, 42, 16, 1 … Σ 1, 3, 19, 87, 451, 2223, 11179, 55767, …
  (−∞, −1]nonnegative
• A116414 → basis A007318 → D+1
  1 · 6, 1 · 32, 12, 1 · 164, 99, 18, 1 … Σ 1, 7, 45, 282, 1737, 10557, 63477, …
  (−∞, 0]nonnegative
• A119308 → reverse
  1 · 1, 2 · 1, 5, 3 · 1, 9, 14, 4 … Σ 1, 3, 9, 28, 90, 297, 1001, 3432, …
  (−∞, 0]nonnegative
• A060920 → Veronese 2 (j=1) → θ+1
  1 · 5 · 20, 2 · 71, 28 … Σ 0, 1, 5, 22, 99, 448, 2013, 8954, …
  (−∞, 0]nonnegative
• A028338 → basis A013610 → θ+1
  1 · 2, 6 · 8, 36, 27 · 48, 264, 324, 108 … Σ 1, 8, 71, 744, 9129, 129072, 2071215, …
  (−∞, 0]nonnegative
• A028326 → D+1
  2 · 4, 2 · 6, 8, 2 · 8, 18, 12, 2 … Σ 2, 6, 16, 40, 96, 224, 512, 1152, …
  (−∞, 0]nonnegativerational roots
• A084534 → θ+1
  1 · 1, 4 · 1, 8, 6 · 1, 12, 27, 8 … Σ 1, 5, 15, 48, 151, 463, 1390, 4105, …
  (−∞, 0]nonnegative
• A013622 → basis A013611 → D
  20 · 320, 800 · 3840, 19200, 24000 … Σ 0, 20, 1120, 47040, 1756160, 61465600, …
  (−∞, 0]nonnegativerational roots
• A079621 → x ↦ x/(x+1)
  1 · 4, 5 · 24, 60, 37 · 192, 720, 888, 361 … Σ 1, 9, 121, 2161, 48081, 1279801, …
  [−1, 0]nonnegative
• A137320 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 6 · 6, 32, 42 · 24, 176, 424, 336 … Σ 1, 8, 80, 960, 13440, 215040, 3870720, …
  [−1, 0]nonnegativerational roots
• A059473 → basis A013609 → θ+1
  1 · 4, 8 · 18, 72, 48 · 80, 480, 672, 256 … Σ 1, 12, 138, 1488, 15444, 156048, …
  (−∞, 0]nonnegative
• A052553 → θ+1
  1 · 1 · 1, 2 · 1, 4 … Σ 1, 1, 3, 5, 10, 18, 33, 59, …
  (−∞, 0]nonnegative
• A008955 → basis A013609 → (θ+1)²
  1 · 2, 8 · 10, 104, 144 · 100, 1760, 5652, 4608 … Σ 1, 10, 258, 12120, 897496, 96114496, …
  (−∞, 0]nonnegative
• A063967 → x ↦ x−2
  1 · -1, 1 · 0, -1, 1 · 1, -1, -1, 1 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (−3, 1]
• A166979 → D
  11, 2 · 111, 60, 3 · 1111, 1362, 174, 4 … Σ 13, 174, 2651, 46511, 926994, 20573827, …
  (−∞, 0]nonnegative
• A144699 → D
  5 · 16, 50 · 39, 342, 375 · 86, 1572, 4578, 2500 … Σ 0, 5, 66, 756, 8736, 105840, 1360800, …
  (−∞, 0]nonnegative
• A145596 → D → θ+1
  2 · 8, 12 · 20, 80, 36 · 40, 300, 360, 80 … Σ 0, 2, 20, 136, 780, 4070, 20020, 94640, …
  (−∞, 0]nonnegative
• A207823 → x ↦ x/(x+1)
  1 · 4, 5 · 15, 38, 24 · 56, 214, 272, 115 … Σ 1, 9, 77, 657, 5605, 47817, 407933, …
  [−1, −2/3)nonnegative
• A126216 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 11, 43, 41 · 45, 261, 495, 306 … Σ 1, 9, 95, 1107, 13757, 178749, 2399251, …
  [−1, 0]nonnegative
• A266213 → (θ+1)²
  1 · 1 · 1, 8 · 1, 16, 18 … Σ 1, 1, 9, 35, 129, 437, 1409, 4375, …
  (−∞, 0]nonnegative
• A136124 → basis A013610 → θ+1
  1 · 3, 6 · 12, 42, 27 · 60, 282, 324, 108 … Σ 1, 9, 81, 774, 7974, 88812, 1067652, …
  (−∞, 0]nonnegative
• A130534 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 6, 29, 46, 24 … Σ 1, 3, 15, 105, 945, 10395, 135135, …
  [−1, 0]nonnegativerational roots
• A038231 → x ↦ x/(x+1)
  1 · 4, 5 · 16, 40, 25 · 64, 240, 300, 125 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  [−1, 0]nonnegativerational roots
• A064861 → x ↦ x/(x+1)
  1, 5, 6 · 1, 8, 21, 18 · 1, 10, 37, 60, 36 … Σ 12, 48, 144, 576, 1728, 6912, 20736, …
  [−1, 0]nonnegativerational roots
• A102413 → basis A007318
  1 · 2, 1 · 6, 6, 1 · 14, 21, 9, 1 … Σ 1, 3, 13, 45, 161, 573, 2041, 7269, …
  (−∞, −1]nonnegative
• A178821 → basis A013609
  1 · 10, 10 · 60, 120, 60 · 280, 840, 840, 280 … Σ 1, 20, 240, 2240, 17920, 129024, …
  (−∞, 0]nonnegativerational roots
• A147716 → basis A013609 → x ↦ x/(x+1)
  1 · 15, 17 · 225, 510, 289 · 3375, 11475, 13005, 4913 … Σ 1, 32, 1024, 32768, 1048576, 33554432, …
  [−1, 0]nonnegativerational roots
• A274496 → D
  2 · 4 · 8, 8 · 16, 16 … Σ 0, 0, 2, 4, 16, 32, 96, 192, …
  (−∞, 0]nonnegativerational roots
• A023531 → x ↦ x−1
  1 · -1, 1 · 1, -2, 1 · -1, 3, -3, 1 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 1][0, ∞)rational roots
• A063967 → Veronese 2 (j=1) → x ↦ x+1
  1 · 3 · 8, 1 · 22, 7 … Σ 0, 1, 3, 9, 29, 92, 292, 927, …
  (−∞, −1]nonnegative
• A038321 → basis A013610 → basis A007318
  1521, 1638, 441 · 59319, 95823, 51597, 9261 … Σ 3600, 216000, 12960000, 777600000, …
  (−∞, −1]nonnegativerational roots
• A063967 → x ↦ x−7
  1 · -6, 1 · 30, -11, 1 · -144, 84, -16, 1 … Σ 1, -5, 20, -75, 275, -1000, 3625, …
  (2, 6]
• A038301 → x ↦ x+1 → x ↦ x/(x+1)
  400, 1240, 961 · 8000, 37200, 57660, 29791 … Σ 2601, 132651, 6765201, 345025251, …
  [−1, 0]nonnegativerational roots
• A371081 → x ↦ x/(x+1)
  1 · 16, 17 · 400, 852, 453 … Σ 1, 33, 1705, 127225, 12963985, …
  [−1, 0]nonnegative
• A082985 → reverse → (θ+1)²
  1 · 3, 4 · 5, 20, 9 · 7, 56, 63, 16 … Σ 1, 7, 34, 142, 541, 1939, 6653, 22091, …
  (−∞, 0]nonnegative
• image under θ+1 → x ↦ x+1
  1 · 2, 2 · 3, 6, 3 · 4, 12, 12, 4 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, −1]nonnegativerational roots
• A305540 → θ+1
  1 · 1, 2 · 1, 4 · 1, 8, 9 … Σ 1, 3, 5, 18, 31, 132, 233, 1152, …
  (−∞, 0]nonnegative
• A330381 → D+1
  1 · 3, 2 · 6, 11, 3 · 10, 35, 25, 4 … Σ 1, 5, 20, 74, 263, 911, 3098, 10388, …
  (−∞, 0]nonnegative
• A038330 → basis A013609 → (θ+1)²
  256, 1024, 576 · 4096, 24576, 27648, 8192 … Σ 1856, 64512, 2101248, 65470464, …
  (−∞, 0]nonnegative
• A056242 → x ↦ x+1 → D+1
  1 · 5, 2 · 23, 21, 4 · 99, 145, 64, 8 … Σ 1, 7, 48, 316, 2019, 12621, 77598, …
  (−∞, 0]nonnegative
• A103327 → basis A013610 → reverse
  1 · 3, 4 · 9, 36, 16 · 27, 216, 240, 64 … Σ 1, 7, 61, 547, 4921, 44287, 398581, …
  (−∞, 0]nonnegative