Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A391403 → x ↦ x+1 → (θ+1)²
  1 · 1 · 3, 8 · 7, 24 … Σ 1, 1, 11, 31, 169, 641, 2891, 11831, …
  (−∞, 0]nonnegative
• A145596 → x ↦ x+1 → θ+1
  1 · 4, 4 · 14, 28, 9 · 48, 144, 96, 16 … Σ 1, 8, 51, 304, 1765, 10128, 57799, …
  (−∞, 0]nonnegative
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
  9, 48, 64 · 27, 216, 576, 512 … Σ 121, 1331, 14641, 161051, 1771561, …
  (−∞, 0)nonnegativerational roots
• delannoy → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 5, 20, 19 · 12, 72, 140, 88 … Σ 1, 6, 44, 312, 2224, 15840, 112832, …
  [−1, 0]nonnegative
• A178821 → basis A013611 → reverse
  1 · 20, 10 · 240, 240, 60 · 2240, 3360, 1680, 280 … Σ 1, 30, 540, 7560, 90720, 979776, …
  (−∞, −1]nonnegativerational roots
• A178822 → Veronese 2 (j=1) → (θ+1)²
  6 · 42 · 168, 224 · 504, 2016 … Σ 0, 6, 42, 392, 2520, 13608, 64680, …
  (−∞, 0]nonnegative
• A102662 → reverse
  1 · 3, 1 · 3, 5, 1 · 3, 11, 7, 1 … Σ 1, 4, 9, 22, 53, 128, 309, 746, …
  (−∞, 0]nonnegative
• A027467 → x ↦ x+1 → θ+1
  1 · 16, 2 · 256, 64, 3 · 4096, 1536, 144, 4 … Σ 1, 18, 323, 5780, 103173, 1837462, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A127647 → D+1
  1 · 2, 1 · 6, 8, 2 · 12, 27, 18, 3 … Σ 1, 3, 16, 60, 240, 896, 3328, 12096, …
  (−∞, 0]nonnegativerational roots
• A172431 → basis A007318 → x ↦ x+1
  1 · 5, 2 · 21, 16, 3 · 85, 94, 34, 4 … Σ 1, 7, 40, 217, 1159, 6160, 32689, …
  (−∞, −1]nonnegative
• A063007 → basis A013610 → θ+1
  1 · 3, 12 · 13, 108, 162 · 63, 792, 2430, 2160 … Σ 1, 15, 283, 5445, 104541, 1995939, …
  (−∞, 0]nonnegative
• A059364 → basis A007318 → basis A007318
  1 · 4, 1 · 28, 15, 2 · 280, 234, 65, 6 … Σ 1, 5, 45, 585, 9945, 208845, 5221125, …
  (−∞, −1]nonnegativerational roots
• A144441 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 16, 16 · 1, 86, 252, 168 … Σ 1, 3, 33, 507, 10537, 278067, 8917809, …
  [−1, 0]nonnegative
• A038335 → x ↦ x+1 → θ+1
  1 · 21, 18 · 441, 756, 243 · 9261, 23814, 15309, 2916 … Σ 1, 39, 1440, 51300, 1782000, 60750000, …
  (−∞, 0]nonnegativerational roots
• A391533 → basis A013609 → D+1
  21, 20, 4 · 60, 106, 56, 8 · 156, 416, 388, 144, 16 … Σ 45, 230, 1120, 5280, 24320, 110080, …
  (−∞, 0]nonnegative
• A113953 → θ+1 → D
  2 · 4, 6 · 0, 24, 12 · 0, 24, 72, 20 … Σ 0, 2, 10, 36, 116, 334, 918, 2400, …
  (−∞, 0]nonnegative
• A144699 → (θ+1)²
  1 · 1, 20 · 1, 64, 225 · 1, 156, 1539, 2000 … Σ 1, 21, 290, 3696, 47460, 633360, …
  (−∞, 0]nonnegative
• A257180 → (θ+1)²
  9, 96, 81 · 27, 564, 1269, 432 … Σ 186, 2292, 29316, 394800, 5624640, …
  (−∞, 0]nonnegative
• A008459 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … Σ 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, 0]nonnegative
• A085478 → reverse → D+1
  1 · 2, 1 · 4, 5, 1 · 6, 17, 9, 1 … Σ 1, 3, 10, 33, 105, 324, 977, 2895, …
  (−∞, 0]nonnegative
• A111006 → D+1
  1 · 1, 1 · 1, 5, 2 · 0, 4, 11, 3 … Σ 1, 2, 8, 18, 48, 110, 260, 586, …
  (−∞, 0]nonnegative
• A257606 → (θ+1)²
  1 · 4, 16 · 16, 160, 144 · 64, 1184, 2664, 1024 … Σ 1, 20, 320, 4936, 77184, 1247040, …
  (−∞, 0]nonnegative
• A159854 → basis A013609 → x ↦ x/(x+1)
  1, 6, 9 · 2, 16, 42, 36 · 4, 40, 148, 240, 144 … Σ 16, 96, 576, 3456, 20736, 124416, …
  [−1, 0]nonnegativerational roots
• A027465 → basis A008277 → x ↦ x+1
  1 · 5, 1 · 26, 11, 1 · 141, 94, 18, 1 … Σ 1, 6, 38, 254, 1790, 13278, 103470, …
  (−∞, −1]nonnegative
• A356546 → basis A013609 → D
  4 · 48, 48 · 480, 960, 480 · 4480, 13440, 13440, 4480 … Σ 0, 4, 96, 1920, 35840, 645120, …
  (−∞, 0]nonnegativerational roots
• A133080 → D → θ+1
  1 · 0, 4 · 0, 4, 9 · 0, 0, 0, 16 … Σ 0, 1, 4, 13, 16, 41, 36, 85, …
  (−∞, 0]nonnegativerational roots
• A001263 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 10 · 1, 15, 45, 35 … Σ 1, 4, 19, 96, 501, 2668, 14407, 78592, …
  [−1, 0]nonnegative
• A144696 → x ↦ x+1
  1 · 3, 2 · 12, 15, 4 · 60, 108, 57, 8 … Σ 1, 5, 31, 233, 2071, 21305, 249271, …
  (−∞, −1]nonnegative
• A038220 → basis A013610 → (θ+1)²
  1 · 5, 24 · 25, 240, 324 · 125, 1800, 4860, 3456 … Σ 1, 29, 589, 10241, 162745, 2437061, …
  (−∞, 0]nonnegative
• A152842 → reverse → D+1
  1 · 4, 1 · 7, 6, 1 · 24, 29, 10, 1 … Σ 1, 5, 14, 64, 160, 704, 1664, 7168, …
  (−∞, 0]nonnegative
• A013620 → basis A013610 → x ↦ x/(x+1)
  1 · 5, 14 · 25, 140, 196 · 125, 1050, 2940, 2744 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  [−1, 0]nonnegativerational roots
• A145597 → (θ+1)²
  1 · 3, 12 · 6, 60, 54 · 10, 180, 405, 160 … Σ 1, 15, 120, 755, 4191, 21567, 105560, …
  (−∞, 0]nonnegative
• A062190 → reverse → x ↦ x+1
  1 · 7, 1 · 36, 16, 1 · 165, 135, 27, 1 … Σ 1, 8, 53, 328, 1966, 11584, 67582, …
  (−∞, −1]nonnegative
• A035342 → θ+1
  1 · 3, 2 · 15, 18, 3 · 105, 174, 54, 4 … Σ 1, 5, 36, 337, 3875, 52776, 830095, …
  (−∞, 0]nonnegative
• A069466 → basis A007318 → θ+1
  1 · 4, 4 · 36, 72, 18 · 400, 1200, 720, 80 … Σ 1, 8, 126, 2400, 49350, 1055376, …
  (−∞, 0]nonnegative
• A110441 → reverse → θ+1
  1 · 1, 6 · 1, 12, 21 · 1, 18, 69, 60 … Σ 1, 7, 34, 148, 612, 2452, 9608, 37024, …
  (−∞, 0]nonnegative
• A038277 → D+1
  1 · 18, 11 · 203, 396, 121 · 1960, 6699, 6534, 1331 … Σ 1, 29, 720, 16524, 361584, 7663248, …
  (−∞, 0]nonnegativerational roots
• A075497 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 14, 11 · 8, 52, 92, 49 … Σ 1, 5, 29, 201, 1657, 15821, 170389, …
  [−1, 0]nonnegative
• A056242 → x ↦ x+1
  1 · 3, 2 · 10, 13, 4 · 34, 65, 40, 8 … Σ 1, 5, 27, 147, 801, 4365, 23787, …
  (−∞, −1]nonnegative
• A241171 → reverse
  1 · 6, 1 · 90, 30, 1 · 2520, 1260, 126, 1 … Σ 1, 7, 121, 3907, 202741, 15430207, …
  (−∞, 0]nonnegative
• A111125 → basis A007318 → θ+1
  1 · 4, 2 · 11, 14, 3 · 29, 62, 30, 4 … Σ 1, 6, 28, 125, 543, 2312, 9694, 40155, …
  (−∞, 0]nonnegative
• A227450 → x ↦ x/(x+1)
  2 · 2, 6 · 2, 16, 22 · 2, 34, 110, 94 … Σ 2, 8, 40, 240, 1664, 12992, 112256, …
  [−1, 0]nonnegative
• A038322 → basis A013610 → D
  24 · 912, 1152 · 25992, 65664, 41472 … Σ 0, 24, 2064, 133128, 7632672, …
  (−∞, 0]nonnegativerational roots
• A094638 → basis A013611 → (θ+1)²
  1 · 2, 16 · 6, 112, 288 · 24, 736, 4176, 6144 … Σ 1, 18, 406, 11080, 355816, 13159376, …
  (−∞, 0]nonnegative
• A069466 → γ → (θ+1)²
  1 · 2 · 6, 48 · 20, 480 … Σ 1, 2, 54, 500, 7210, 88452, 1155924, …
  (−∞, 0]nonnegative
• A347056 → D
  4 · 16, 20 · 40, 100, 60 · 80, 300, 360, 140 … Σ 0, 4, 36, 200, 880, 3360, 11648, 37632, …
  (−∞, 0]nonnegativerational roots
• A059473 → basis A013609 → reverse
  1 · 4, 4 · 16, 36, 18 · 64, 224, 240, 80 … Σ 1, 8, 70, 608, 5284, 45920, 399064, …
  (−∞, 0]nonnegative
• A142459 → (θ+1)²
  1 · 1, 4 · 1, 40, 9 · 1, 236, 531, 16 … Σ 1, 5, 50, 784, 15744, 384240, 11004000, …
  (−∞, 0]nonnegative
• A118933 → basis A013610 → x ↦ x+1
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 43, 211, 631, 1471, …
  (−∞, −1]nonnegative
• A172431 → (θ+1)²
  1 · 1, 8 · 1, 16, 27 · 1, 24, 90, 64 … Σ 1, 9, 44, 179, 667, 2352, 7973, 26229, …
  (−∞, 0]nonnegative