Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A103406 → Veronese 2 → D+1
  1 · 1 · 4, 3 · 10, 9 … Σ 1, 1, 7, 19, 52, 136, 344, 848, …
  (−∞, 0]nonnegative
• A116666 → x ↦ x+1
  1 · 4, 3 · 12, 16, 5 · 32, 60, 36, 7 … Σ 1, 7, 33, 135, 513, 1863, 6561, 22599, …
  (−∞, −1]nonnegativerational roots
• A038328 → x ↦ x+1 → x ↦ x+1
  256, 64, 4 · 4096, 1536, 192, 8 … Σ 324, 5832, 104976, 1889568, 34012224, …
  (−∞, −1]nonnegativerational roots
• A086645 → γ → x ↦ x+1
  1 · 1 · 5, 4 · 13, 12 … Σ 1, 1, 9, 25, 113, 401, 1593, 5993, …
  (−∞, −1]nonnegative
• A038328 → D+1
  192, 56, 4 · 2592, 1152, 168, 8 … Σ 252, 3920, 60368, 921984, 13983424, …
  (−∞, 0]nonnegativerational roots
• A113214 → basis A013610 → D
  3 · 15, 18 · 39, 126, 81 · 87, 468, 729, 324 … Σ 0, 3, 33, 246, 1608, 9792, 57072, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
  1 · -4, -3 · 16, 24, 9 · -64, -144, -108, -27 … Σ 1, -7, 49, -343, 2401, -16807, 117649, …
  (−∞, −1]rational roots
• A099040 → basis A013610 → (θ+1)²
  1 · 2, 24 · 6, 120, 324 · 16, 480, 2592, 3456 … Σ 1, 26, 450, 6544, 86120, 1061568, …
  (−∞, 0]nonnegative
• A133087 → x ↦ x+1 → x ↦ x+1
  1 · 4, 1 · 9, 6, 1 · 36, 33, 10, 1 … Σ 1, 5, 16, 80, 256, 1280, 4096, 20480, …
  (−∞, −1]nonnegativerational roots
• A038226 → basis A007318 → D
  176, 128 · 2904, 4224, 1536 … Σ 304, 8664, 219488, 5212840, 118852752, …
  (−∞, 0]nonnegativerational roots
• A059364 → basis A013610 → D+1
  1 · 6, 3 · 48, 69, 18 · 471, 1212, 909, 162 … Σ 1, 9, 135, 2754, 70506, 2165796, …
  (−∞, 0]nonnegative
• A085478 → reverse
  1 · 1, 1 · 1, 3, 1 · 1, 5, 6, 1 … Σ 1, 2, 5, 13, 34, 89, 233, 610, …
  (−∞, 0]nonnegative
• A035342 → x ↦ x+1
  1 · 4, 1 · 25, 11, 1 · 211, 126, 21, 1 … Σ 1, 5, 37, 359, 4291, 60797, 994645, …
  (−∞, −1]nonnegative
• A132440 → θ+1 → (θ+1)²
  1 · 0, 16 · 0, 0, 81 · 0, 0, 0, 256 … Σ 0, 1, 16, 81, 256, 625, 1296, 2401, …
  (−∞, 0]nonnegativerational roots
• A038338 → basis A013610 → reverse
  1296, 1728, 576 · 46656, 93312, 62208, 13824 … Σ 3600, 216000, 12960000, 777600000, …
  (−∞, 0]nonnegativerational roots
• A136572 → D → D+1
  1 · 4, 4 · 0, 36, 18 · 0, 0, 288, 96 … Σ 0, 1, 8, 54, 384, 3000, 25920, 246960, …
  (−∞, 0]nonnegativerational roots
• A034867 → D
  1 · 4 · 10, 2 · 20, 12 … Σ 0, 0, 1, 4, 12, 32, 80, 192, …
  (−∞, 0]nonnegative
• A038329 → basis A013609 → θ+1
  225, 360, 108 · 3375, 8100, 4860, 864 … Σ 693, 17199, 416745, 9918531, 232793757, …
  (−∞, 0]nonnegativerational roots
• A225468 → (θ+1)²
  1 · 2, 4 · 4, 28, 9 · 8, 156, 135, 16 … Σ 1, 6, 41, 315, 2700, 25587, 265437, …
  (−∞, 0]nonnegative
• A081578: Pascal-(1,3,1) array.
  1 · 1, 1 · 1, 5, 1 · 1, 9, 9, 1 … Σ 1, 2, 7, 20, 61, 182, 547, 1640, …
  (−∞, 0)nonnegative
• A390433 → D+1
  1 · 1, 1 · 4, 6, 1 · 36, 76, 23, 1 … Σ 1, 2, 11, 136, 2989, 102142, 4986679, …
  (−∞, 0]nonnegative
• A290315 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 18, 29 · 1, 69, 419, 471 … Σ 1, 4, 48, 960, 26880, 967680, 42577920, …
  [−1, 0]nonnegative
• A178821 → (θ+1)²
  1 · 5, 20 · 15, 120, 135 · 35, 420, 945, 560 … Σ 1, 25, 270, 1960, 11200, 54432, 235200, …
  (−∞, 0]nonnegative
• A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−1, 0)nonnegative
• A178822 → basis A013610 → θ+1
  1 · 12, 36 · 84, 504, 567 · 448, 4032, 9072, 6048 … Σ 1, 48, 1155, 19600, 267750, 3150000, …
  (−∞, 0]nonnegativerational roots
• A249248 → D+1
  1 · 4, 3 · 8, 13, 3 · 13, 56, 31, 3 … Σ 1, 7, 24, 103, 419, 1815, 7970, 36277, …
  (−∞, 0]nonnegative
• A094816 → D+1
  1 · 2, 1 · 4, 5, 1 · 9, 20, 9, 1 … Σ 1, 3, 10, 39, 181, 995, 6386, 47075, …
  (−∞, 0]nonnegative
• A196347 → basis A013609
  1 · 2, 2 · 8, 16, 8 · 48, 144, 144, 48 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
  (−∞, 0]nonnegativerational roots
• A011973 → basis A013611 → basis A007318
  1 · 1 · 6, 4 · 11, 8 … Σ 1, 1, 10, 19, 109, 280, 1261, 3781, …
  (−∞, −1]nonnegative
• A039756 → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 13, 9, 1 … Σ 1, 2, 6, 24, 116, 648, 4088, 28640, …
  (−∞, 0]nonnegative
• A104978 → D
  1 · 5, 6 · 21, 56, 36 · 84, 360, 495, 220 … Σ 0, 1, 11, 113, 1159, 11969, 124515, …
  (−∞, 0]nonnegative
• A093375 → Veronese 2 (j=1) → x ↦ x+1
  1 · 4 · 10, 1 · 24, 8 … Σ 0, 1, 4, 11, 32, 89, 244, 659, …
  (−∞, −1]nonnegative
• A008278 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 10, 23, 15 … Σ 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, 0]nonnegative
• A109983 → basis A007318
  1 · 3, 5, 2 · 13, 44, 55, 30, 6 … Σ 1, 10, 148, 2440, 42256, 752800, …
  (−∞, −1]nonnegative
• A011973 → basis A013610 → basis A007318
  1 · 1 · 5, 3 · 9, 6 … Σ 1, 1, 8, 15, 71, 176, 673, 1905, …
  (−∞, −1]nonnegative
• A008277 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 15, 67, 98, 47 … Σ 1, 5, 31, 227, 1897, 17693, 181351, …
  [−1, 0]nonnegative
• A084534 → basis A007318
  1 · 3, 2 · 7, 8, 2 · 18, 30, 15, 2 … Σ 1, 5, 17, 65, 257, 1025, 4097, 16385, …
  (−∞, −1]nonnegative
• A008277 → D
  1 · 3, 2 · 7, 12, 3 · 15, 50, 30, 4 … Σ 0, 1, 5, 22, 99, 471, 2386, 12867, …
  (−∞, 0]nonnegative
• A013622 → basis A013611 → (θ+1)²
  1 · 8, 80 · 64, 1280, 3600 … Σ 1, 88, 4944, 230272, 9646336, …
  (−∞, 0]nonnegative
• A125092 → D+1 → θ+1
  1 · 5, 8 · 9, 52, 27 · 13, 132, 225, 64 … Σ 1, 13, 88, 434, 1784, 6512, 21856, …
  (−∞, 0]nonnegative
• chebyshev_u → x ↦ x+1 → D
  2 · 8, 8 · 20, 48, 24 · 40, 168, 192, 64 … Σ 0, 2, 16, 92, 464, 2182, 9824, 42936, …
  (−∞, 0]nonnegative
• A038301 → basis A013609 → θ+1
  400, 1760, 1452 · 8000, 52800, 87120, 42592 … Σ 3612, 190512, 9631440, 472977792, …
  (−∞, 0]nonnegativerational roots
• A091042 → basis A007318 → D
  3 · 20, 10 · 112, 112, 21 · 576, 864, 360, 36 … Σ 0, 3, 30, 245, 1836, 13079, 90090, …
  (−∞, 0]nonnegative
• A125662 → basis A013609 → (θ+1)²
  1 · 4, 8 · 15, 64, 36 · 56, 368, 432, 128 … Σ 1, 12, 115, 984, 7861, 59940, 441767, …
  (−∞, 0]nonnegative
• A028326 → D → D
  4 · 12, 12 · 24, 48, 24 · 40, 120, 120, 40 … Σ 0, 0, 4, 24, 96, 320, 960, 2688, …
  (−∞, 0]nonnegativerational roots
• A086645 → basis A013609 → D
  2 · 16, 8 · 96, 144, 24 · 512, 1280, 768, 64 … Σ 0, 2, 24, 264, 2624, 24480, 219264, …
  (−∞, 0]nonnegative
• A201730 → D+1
  1 · 2 · 6, 1 · 20, 6 … Σ 1, 2, 7, 26, 96, 352, 1284, 4664, …
  (−∞, 0]nonnegative
• A108838 → (θ+1)²
  2 · 3, 8 · 4, 32, 18 · 5, 80, 135, 32 … Σ 2, 11, 54, 252, 1140, 5049, 22022, …
  (−∞, 0]nonnegative
• A136572 → Veronese 2 (j=1) → x ↦ x+1
  1 · 6, 6 · 120, 240, 120 · 5040, 15120, 15120, 5040 … Σ 0, 1, 0, 12, 0, 480, 0, 40320, …
  (−∞, −1]nonnegativerational roots
• A078812 → basis A013610 → θ+1
  1 · 3, 6 · 8, 36, 27 · 21, 150, 243, 108 … Σ 1, 9, 71, 522, 3673, 25083, 167615, …
  (−∞, 0]nonnegative