Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A110552 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 13, 40, 36 … Σ 1, 4, 18, 90, 486, 2754, 16038, 94770, …
  [−1, 0]nonnegativerational roots
• A136394 → basis A007318
  1 · 1 · 2, 1 · 6, 5 … Σ 1, 1, 3, 11, 53, 309, 2119, 16687, …
  (−∞, −1]nonnegative
• A111125 → basis A013610 → θ+1
  1 · 4, 6 · 11, 42, 27 · 29, 186, 270, 108 … Σ 1, 10, 80, 593, 4195, 28756, 192698, …
  (−∞, 0]nonnegative
• A075503 → x ↦ x+1
  89, 26, 1 · 1009, 547, 51, 1 … Σ 116, 1608, 26512, 506272, 10904896, …
  (−∞, −1]nonnegative
• A035469 → θ+1
  1 · 4, 2 · 28, 24, 3 · 280, 320, 72, 4 … Σ 0, 1, 6, 55, 676, 10405, 192266, …
  (−∞, 0]nonnegative
• A085881 → basis A007318 → (θ+1)²
  1 · 2, 4 · 12, 48, 27 · 120, 720, 810, 240 … Σ 1, 6, 87, 1890, 53865, 1888110, …
  (−∞, 0]nonnegative
• A085881 → θ+1
  1 · 1, 2 · 3, 12, 9 · 15, 90, 135, 60 … Σ 1, 3, 24, 300, 5040, 105840, 2661120, …
  (−∞, 0]nonnegativerational roots
• A081582 → γ → basis A007318
  1 · 1 · 8, 7 · 15, 14 … Σ 1, 1, 15, 29, 239, 645, 3991, 13021, …
  (−∞, −1]nonnegative
• A074246 → D+1
  1 · 5, 2 · 23, 18, 3 · 120, 130, 42, 4 … Σ 1, 7, 44, 296, 2204, 18236, 166984, …
  (−∞, 0]nonnegative
• image under θ+1 → (θ+1)² → reverse
  1 · 8 · 27 · 64 … Σ 1, 8, 27, 64, 125, 216, 343, 512, …
  (−∞, 0]nonnegativerational roots
• A136572 → basis A013610 → D
  3 · 12, 36 · 54, 324, 486 · 288, 2592, 7776, 7776 … Σ 0, 3, 48, 864, 18432, 460800, 13271040, …
  (−∞, 0]nonnegativerational roots
• A147716 → basis A013611 → (θ+1)²
  1 · 15, 16 · 225, 480, 144 · 3375, 10800, 6480, 1024 … Σ 1, 31, 849, 21679, 528865, 12490239, …
  (−∞, 0]nonnegative
• A125662 → (θ+1)²
  1 · 3, 4 · 8, 24, 9 · 21, 100, 81, 16 … Σ 1, 7, 41, 218, 1091, 5237, 24379, …
  (−∞, 0]nonnegative
• Veronese parent of A109447 → D+1
  1, 3, 1 · 3, 4, 1 · 1, 7, 12, 7, 1 · 5, 16, 18, 8, 1 … Σ 5, 8, 28, 48, 144, 256, 704, 1280, …
  (−∞, 0]nonnegative
• A099040 → D+1
  1 · 2, 2 · 2, 10, 4 · 0, 16, 32, 8 … Σ 1, 4, 16, 56, 188, 608, 1920, 5952, …
  (−∞, 0]nonnegative
• A249248: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 7, 3 · 1, 12, 22, 3 … Σ 1, 4, 11, 38, 131, 490, 1897, 7714, …
  (−∞, 0)nonnegative
• A142462 → θ+1
  1 · 1, 2 · 1, 32, 3 · 1, 286, 429, 4 … Σ 1, 3, 36, 720, 19872, 695520, 29410560, …
  (−∞, 0]nonnegative
• A038303 → basis A007318 → D
  1 · 22, 2 · 363, 66, 3 · 5324, 1452, 132, 4 … Σ 0, 1, 24, 432, 6912, 103680, 1492992, …
  (−∞, 0]nonnegativerational roots
• A038303 → basis A013609 → (θ+1)²
  1 · 11, 8 · 121, 176, 36 · 1331, 2904, 1188, 128 … Σ 1, 19, 333, 5551, 89401, 1403883, …
  (−∞, 0]nonnegative
• A013611 → basis A007318 → D+1
  1 · 9, 4 · 65, 72, 16 · 425, 780, 432, 64 … Σ 1, 13, 153, 1701, 18225, 190269, …
  (−∞, 0]nonnegativerational roots
• chebyshev_t → x ↦ x+1
  1 · 1, 1 · 1, 4, 2 · 1, 9, 12, 4 … Σ 1, 2, 7, 26, 97, 362, 1351, 5042, …
  (−2, 0)nonnegative
• A114687 → basis A007318 → x ↦ x/(x+1)
  1 · 3, 5 · 11, 36, 29 · 45, 219, 351, 185 … Σ 1, 8, 76, 800, 8992, 105728, 1284544, …
  [−1, 0]nonnegative
• A038208 → basis A013610 → D
  6 · 48, 72 · 288, 864, 648 · 1536, 6912, 10368, 5184 … Σ 0, 6, 120, 1800, 24000, 300000, …
  (−∞, 0]nonnegativerational roots
• A143495 → x ↦ x/(x+1)
  1 · 3, 4 · 9, 25, 17 · 27, 118, 167, 77 … Σ 1, 7, 51, 389, 3121, 26423, 236403, …
  [−1, 0]nonnegative
• basis-map coefficients of A scaled Hermite triangle.
  0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 0, 1 … Σ 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0]nonnegativerational roots
• A062145 → reverse → x ↦ x/(x+1)
  1 · 4, 5 · 10, 30, 21 · 20, 105, 168, 84 … Σ 1, 9, 61, 377, 2241, 13073, 75517, …
  [−1, 0]nonnegative
• basis-map coefficients of A371080, shifted → basis A013610 → (θ+1)²
  1 · 1, 12 · 4, 60, 81 · 28, 468, 972, 432 … Σ 1, 13, 145, 1900, 29704, 545356, …
  (−∞, 0]nonnegative
• A200536 → basis A013609 → D+1
  1 · 20, 30, 8 · 204, 752, 964, 480, 64 … Σ 1, 58, 2464, 92512, 3248896, 109408768, …
  (−∞, 0]nonnegative
• A091044 → basis A013610 → D
  6 · 48, 54 · 288, 720, 324 · 1536, 6048, 6480, 1620 … Σ 0, 6, 102, 1332, 15684, 174378, …
  (−∞, 0]nonnegative
• A038231 → Veronese 2 → D+1
  1 · 4 · 17, 1 · 76, 12 … Σ 1, 4, 18, 88, 451, 2364, 12500, 66224, …
  (−∞, 0]nonnegative
• A132393 → basis A008277 → θ+1
  1 · 1, 2 · 2, 8, 3 · 6, 36, 27, 4 … Σ 1, 3, 13, 73, 501, 4051, 37633, 394353, …
  (−∞, 0]nonnegative
• A085881 → basis A013611 → reverse
  1 · 4, 2 · 48, 48, 12 · 960, 1440, 720, 120 … Σ 1, 6, 108, 3240, 136080, 7348320, …
  (−∞, −1]nonnegativerational roots
• A243662 → D
  1 · 8, 2 · 55, 30, 3 · 364, 312, 72, 4 … Σ 0, 1, 10, 88, 752, 6375, 53966, 457240, …
  (−∞, 0]nonnegative
• A049403 → x ↦ x/(x+1)
  0, 3, 4 · 0, 3, 12, 10 · 0, 0, 15, 40, 26 … Σ 7, 25, 81, 331, 1303, 5937, 26785, …
  [−1, 0]nonnegative
• A052553 → x ↦ x+1 → θ+1
  1 · 1 · 2, 2 · 3, 4 … Σ 1, 1, 4, 7, 18, 37, 84, 179, …
  (−∞, 0]nonnegative
• A136572 → θ+1 → D
  2 · 0, 12 · 0, 0, 72 · 0, 0, 0, 480 … Σ 0, 2, 12, 72, 480, 3600, 30240, 282240, …
  (−∞, 0]nonnegativerational roots
• A019538 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 13 · 1, 17, 67, 75 … Σ 1, 4, 22, 160, 1456, 15904, 202672, …
  [−1, 0)nonnegative
• A145324 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 9 · 12, 75, 117 · 60, 579, 1860, 1989 … Σ 1, 12, 204, 4488, 121176, 3877632, …
  [−1, 0]nonnegativerational roots
• A113953 → basis A013609 → x ↦ x/(x+1)
  1 · 1, 3 · 3, 14, 15 · 5, 37, 87, 63 … Σ 1, 4, 32, 192, 1280, 8192, 53248, …
  [−1, 0]nonnegative
• A113214 → basis A007318 → D+1
  1 · 4, 1 · 9, 7, 1 · 20, 27, 10, 1 … Σ 1, 5, 17, 58, 188, 596, 1852, 5672, …
  (−∞, 0]nonnegative
• A249139 → basis A007318 → D+1
  1 · 5, 1 · 9, 2 · 28, 11, 1 … Σ 1, 6, 11, 40, 93, 282, 719, 2028, …
  (−∞, 0]nonnegative
• A023531 → Veronese 2 → (θ+1)²
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … Σ 1, 0, 4, 0, 9, 0, 16, 0, …
  (−∞, 0]nonnegativerational roots
• A049403 → θ+1
  0, 6, 3 · 0, 6, 18, 4 · 0, 0, 45, 40, 5 … Σ 9, 28, 90, 306, 1078, 3984, 15228, …
  (−∞, 0]nonnegative
• A130850 → x ↦ x+1
  1 · 2, 1 · 6, 5, 1 · 26, 29, 10, 1 … Σ 1, 3, 12, 66, 480, 4368, 47712, 608016, …
  (−∞, −1]nonnegative
• A272867 → basis A013611 → x ↦ x+1
  1 · 46, 56, 16 · 2116, 5152, 4608, 1792, 256 … Σ 1, 118, 13924, 1643032, 193877776, …
  (−∞, −1]nonnegative
• A161198 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 9 · 15, 78, 99 · 105, 741, 1707, 1287 … Σ 1, 12, 192, 3840, 92160, 2580480, …
  [−1, 0]nonnegativerational roots
• A049029 → D+1
  1 · 6, 1 · 60, 17, 1 · 840, 315, 33, 1 … Σ 1, 7, 78, 1189, 22975, 537366, …
  (−∞, 0]nonnegative
• A147716 → basis A013611
  1 · 15, 4 · 225, 120, 16 · 3375, 2700, 720, 64 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  (−∞, −1]nonnegativerational roots
• A181371 → D
  1 · 6 · 25, 2 · 90, 18 … Σ 0, 0, 1, 6, 27, 108, 405, 1458, …
  (−∞, 0]nonnegative
• A357613 → (θ+1)²
  1 · 3, 8 · 15, 80, 54 · 84, 672, 945, 320 … Σ 1, 11, 149, 2021, 26929, 352415, …
  (−∞, 0]nonnegative