Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A013622 → x ↦ x/(x+1) → θ+1
  1 · 3, 16 · 9, 96, 192 · 27, 432, 1728, 2048 … Σ 1, 19, 297, 4235, 57233, 746691, …
  (−∞, 0]nonnegativerational roots
• A290596 → θ+1
  1 · 2, 2 · 10, 20, 3 · 80, 240, 72, 4 … Σ 1, 4, 33, 396, 6165, 117396, 2636469, …
  (−∞, 0]nonnegative
• A181371 → θ+1
  1 · 3 · 8, 2 · 21, 12 … Σ 1, 3, 10, 33, 108, 351, 1134, 3645, …
  (−∞, 0]nonnegative
• A038279 → reverse
  1 · 1, 8 · 1, 16, 64 · 1, 24, 192, 512 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, 0]nonnegativerational roots
• A062196: Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
  1 · 1, 3 · 1, 8, 6 · 1, 15, 30, 10 … Σ 1, 4, 15, 56, 210, 792, 3003, 11440, …
  (−∞, 0)nonnegative
• A123160 → D
  1 · 4, 6 · 9, 36, 30 · 16, 120, 240, 140 … Σ 0, 1, 10, 75, 516, 3405, 21918, 138775, …
  (−∞, 0]nonnegative
• A207607 → D+1
  1 · 3, 2 · 6, 9, 2 · 10, 27, 15, 2 … Σ 1, 5, 17, 54, 166, 499, 1475, 4303, …
  (−∞, 0]nonnegative
• A143495 → x ↦ x+1 → D+1
  1 · 5, 1 · 26, 11, 1 · 141, 94, 18, 1 … Σ 1, 6, 38, 254, 1790, 13278, 103470, …
  (−∞, −1]nonnegative
• A156365 → x ↦ x/(x+1)
  1, 10, 13 · 1, 25, 91, 75 · 1, 56, 426, 896, 541 … Σ 24, 192, 1920, 23040, 322560, 5160960, …
  [−1, 0]nonnegative
• A169803 → θ+1
  1 · 1, 2 · 1, 4 · 1, 6, 3 … Σ 1, 3, 5, 10, 18, 33, 59, 105, …
  (−∞, 0]nonnegative
• A038329 → D+1
  216, 90, 9 · 3024, 1944, 405, 27 … Σ 315, 5400, 91125, 1518750, 25059375, …
  (−∞, 0]nonnegativerational roots
• Veronese parent of A109446 → Veronese 2 (j=1) → x ↦ x+1
  1 · 1 · 2, 1 · 4, 1 … Σ 1, 1, 3, 5, 17, 29, 99, 169, …
  (−∞, −1]nonnegative
• A110441 → basis A013611 → D+1
  1 · 8, 4 · 46, 64, 16 · 224, 560, 384, 64 … Σ 1, 12, 126, 1232, 11524, 104592, …
  (−∞, 0]nonnegative
• A136124 → basis A008277 → θ+1
  1 · 3, 2 · 12, 16, 3 · 60, 120, 45, 4 … Σ 1, 5, 31, 229, 1961, 19081, 207775, …
  (−∞, 0]nonnegative
• A063967 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 6, 27, 30 · 16, 108, 240, 175 … Σ 1, 7, 63, 539, 4655, 40131, 346087, …
  [−1, 0]nonnegative
• A038305 → basis A013610 → reverse
  81, 234, 169 · 729, 3159, 4563, 2197 … Σ 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0]nonnegativerational roots
• A137320 → Veronese 2 (j=1) → D+1
  2 · 2 · 12, 8 · 60, 48 … Σ 0, 2, 2, 20, 108, 704, 5280, 44736, …
  (−∞, 0]nonnegative
• A122076 → x ↦ x/(x+1)
  2 · 3, 5 · 7, 22, 17 · 18, 84, 129, 65 … Σ 2, 8, 46, 296, 1954, 12968, 86158, …
  [−1, 0]nonnegative
• A120909 → Veronese 2 (j=1) → D+1
  6 · 12 · 42, 24 · 120, 96 … Σ 0, 6, 12, 66, 216, 798, 2724, 9306, …
  (−∞, 0]nonnegative
• A136124 → basis A013611 → x ↦ x/(x+1)
  1 · 3, 7 · 12, 52, 56 · 60, 368, 748, 504 … Σ 1, 10, 120, 1680, 26880, 483840, …
  [−1, 0]nonnegativerational roots
• A046716 → basis A013609 → D
  2 · 10, 8 · 50, 88, 24 · 288, 856, 672, 64 … Σ 0, 2, 18, 162, 1880, 28794, 546282, …
  (−∞, 0]nonnegative
• A207823 → basis A013610 → D
  3 · 30, 18 · 219, 270, 81 · 1410, 2646, 1620, 324 … Σ 0, 3, 48, 570, 6000, 59145, 559392, …
  (−∞, 0]nonnegative
• A098158 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 0, 3, 4 … Σ 1, 1, 3, 7, 17, 41, 99, 239, …
  [−1, 0]nonnegative
• A204420 → basis A013609 → reverse
  1 · 2, 1 · 12, 24, 9 · 120, 540, 690, 225 … Σ 1, 3, 45, 1575, 99225, 9823275, …
  (−∞, 0]nonnegativerational roots
• A143685 → basis A007318 → θ+1
  1 · 2, 2 · 13, 26, 3 · 44, 132, 72, 4 … Σ 1, 4, 42, 252, 1890, 12312, 84078, …
  (−∞, 0]nonnegative
• A028326 → basis A013609 → (θ+1)²
  2 · 4, 16 · 8, 64, 72 · 16, 192, 432, 256 … Σ 2, 20, 144, 896, 5120, 27648, 143360, …
  (−∞, 0]nonnegative
• A330510 → D+1
  2 · 6, 3 · 14, 18, 4 · 27, 66, 37, 5 … Σ 2, 9, 36, 135, 486, 1701, 5832, 19683, …
  (−∞, 0]nonnegative
• A139548 → x ↦ x+1
  1 · 8, 8, 2 · 64, 128, 96, 32, 4 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, −1]nonnegativerational roots
• A091042 → D
  3 · 10, 10 · 21, 70, 21 · 36, 252, 252, 36 … Σ 0, 3, 20, 112, 576, 2816, 13312, 61440, …
  (−∞, 0]nonnegative
• A143494 → x ↦ x+1 → x ↦ x+1
  1 · 4, 1 · 18, 9, 1 · 90, 67, 15, 1 … Σ 1, 5, 28, 173, 1165, 8468, 65923, …
  (−∞, −1]nonnegative
• A013609 → basis A008288 → (θ+1)²
  1 · 3, 8 · 9, 64, 36 · 27, 328, 468, 128 … Σ 1, 11, 109, 951, 7721, 59667, 444725, …
  (−∞, 0]nonnegative
• A144441 → D+1
  1 · 2, 1 · 15, 16, 1 · 84, 249, 86, 1 … Σ 1, 3, 32, 420, 7248, 154560, 3924736, …
  (−∞, 0]nonnegative
• A038324 → D
  10 · 220, 200 · 3630, 6600, 3000 … Σ 0, 10, 420, 13230, 370440, 9724050, …
  (−∞, 0]nonnegativerational roots
• A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the…
  1 · 1, 1 · 1, 3, 2 · 1, 6, 11, 6 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0)nonnegativerational roots
• narayana → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … Σ 1, 3, 11, 45, 197, 903, 4279, 20793, …
  [−1, 0]nonnegative
• basis-map coefficients of A112227 → θ+1 → reverse
  3 · 4 · 5 · 6 … Σ 3, 4, 5, 6, 7, 8, 9, 10, …
  (−∞, 0]nonnegativerational roots
• A078812 → x ↦ x/(x+1)
  1 · 2, 3 · 3, 10, 8 · 4, 22, 38, 21 … Σ 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  [−1, 0)nonnegative
• A225471 → basis A013610 → basis A007318
  1 · 7, 3 · 77, 54, 9 · 1155, 1041, 297, 27 … Σ 1, 10, 140, 2520, 55440, 1441440, …
  (−∞, −1]nonnegativerational roots
• A116414 → θ+1
  1 · 4, 2 · 13, 16, 3 · 40, 84, 36, 4 … Σ 1, 6, 32, 164, 819, 4012, 19363, 92352, …
  (−∞, 0]nonnegative
• A139548 → basis A013609 → basis A007318
  1 · 32, 32, 8 · 1024, 2048, 1536, 512, 64 … Σ 1, 72, 5184, 373248, 26873856, …
  (−∞, −1]nonnegativerational roots
• A304236 → basis A007318
  1 · 1 · 4, 3 · 7, 6 … Σ 1, 1, 7, 13, 55, 133, 463, 1261, …
  (−∞, −1]nonnegative
• A038319 → basis A013610 → θ+1
  256, 960, 675 · 4096, 23040, 32400, 13500 … Σ 1891, 73036, 2710981, 97893226, …
  (−∞, 0]nonnegativerational roots
• A106246 → θ+1 → x ↦ x+1
  1 · 4, 2 · 12, 14, 3 · 28, 54, 30, 4 … Σ 1, 6, 29, 116, 408, 1312, 3952, 11328, …
  (−∞, −1]nonnegative
• A352880 → reverse
  8 · 18, 12 · 32, 64, 16 · 50, 200, 150, 20 … Σ 8, 30, 112, 420, 1584, 6006, 22880, …
  (−∞, 0]nonnegative
• A125092 → D+1 → x ↦ x/(x+1)
  1 · 5, 9 · 9, 44, 44 · 13, 105, 246, 170 … Σ 1, 14, 97, 534, 2601, 11718, 49977, …
  [−1, 0]nonnegative
• A048993 → θ+1 → reverse
  1 · 2 · 3, 2 · 4, 9, 2 … Σ 1, 2, 5, 15, 52, 203, 877, 4140, …
  (−∞, 0]nonnegative
• A356546 → basis A013611
  1 · 4, 8 · 24, 96, 96 · 160, 960, 1920, 1280 … Σ 1, 12, 216, 4320, 90720, 1959552, …
  [−1, 0]nonnegativerational roots
• A081582 → γ → D+1
  1 · 1 · 8, 7 · 15, 14 … Σ 1, 1, 15, 29, 190, 498, 2325, 7043, …
  (−∞, 0]nonnegative
• A026374 → D
  1 · 3, 2 · 4, 8, 3 · 6, 22, 18, 4 … Σ 0, 1, 5, 15, 50, 125, 375, 875, …
  (−∞, 0]nonnegative
• A038220 → basis A013610 → D
  6 · 60, 72 · 450, 1080, 648 … Σ 0, 6, 132, 2178, 31944, 439230, …
  (−∞, 0]nonnegativerational roots