Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038319 → x ↦ x+1 → x ↦ x/(x+1)
  256, 672, 441 · 4096, 16128, 21168, 9261 … Σ 1369, 50653, 1874161, 69343957, …
  [−1, 0]nonnegativerational roots
• A038325 → basis A007318 → D
  11 · 484, 242 · 15972, 15972, 3993 … Σ 0, 11, 726, 35937, 1581228, 65225655, …
  (−∞, 0]nonnegativerational roots
• A330798 → x ↦ x/(x+1)
  1 · 2, 4 · 6, 27, 30 · 20, 144, 340, 264 … Σ 1, 6, 63, 768, 10005, 135234, 1870932, …
  [−1, 0]nonnegative
• A027465 → D → θ+1
  1 · 6, 4 · 27, 36, 9 · 108, 216, 108, 16 … Σ 0, 1, 10, 72, 448, 2560, 13824, 71680, …
  (−∞, 0]nonnegativerational roots
• A008459 → basis A013609 → θ+1
  1 · 2, 4 · 6, 24, 12 · 20, 120, 144, 32 … Σ 1, 6, 42, 316, 2430, 18804, 145684, …
  (−∞, 0]nonnegative
• A225471 → basis A008277 → D
  1 · 13, 2 · 201, 54, 3 · 3801, 1374, 138, 4 … Σ 0, 1, 15, 258, 5317, 129315, 3640926, …
  (−∞, 0]nonnegative
• A091042 → reverse → x ↦ x/(x+1)
  1 · 3, 4 · 5, 20, 16 · 7, 56, 112, 64 … Σ 1, 7, 41, 239, 1393, 8119, 47321, …
  [−1, 0]nonnegative
• A008459: Square the entries of Pascal's triangle.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 9, 1 … Σ 1, 2, 6, 20, 70, 252, 924, 3432, …
  (−∞, 0)nonnegative
• A136572 → basis A013609 → D
  2 · 8, 16 · 36, 144, 144 · 192, 1152, 2304, 1536 … Σ 0, 2, 24, 324, 5184, 97200, 2099520, …
  (−∞, 0]nonnegativerational roots
• A257606 → θ+1
  1 · 4, 8 · 16, 80, 48 · 64, 592, 888, 256 … Σ 1, 12, 144, 1800, 23760, 332640, …
  (−∞, 0]nonnegative
• A290319 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 8, 12 · 1, 18, 92, 120 … Σ 1, 3, 21, 231, 3465, 65835, 1514205, …
  (−∞, 0]nonnegativerational roots
• A123519 → basis A007318 → reverse
  4, 14, 11 · 8, 44, 76, 41 · 16, 120, 324, 372, 153 … Σ 29, 169, 985, 5741, 33461, 195025, …
  (−1, 0]nonnegative
• A038327 → basis A013609 → x ↦ x/(x+1)
  1 · 13, 15 · 169, 390, 225 · 2197, 7605, 8775, 3375 … Σ 1, 28, 784, 21952, 614656, 17210368, …
  [−1, 0]nonnegativerational roots
• A116414 → basis A013610 → D+1
  1 · 8, 3 · 52, 48, 9 · 302, 477, 216, 27 … Σ 1, 11, 109, 1022, 9241, 81449, 704389, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → (θ+1)²
  1 · 0, -4 · 0, 0, 9 · 0, 0, 0, -16 … Σ 1, -4, 9, -16, 25, -36, 49, -64, …
  [−1, 0]rational roots
• A038329 → x ↦ x/(x+1)
  144, 360, 225 · 1728, 6480, 8100, 3375 … Σ 729, 19683, 531441, 14348907, …
  [−1, 0]nonnegativerational roots
• A166343 → (θ+1)²
  1, 48, 9 · 1, 108, 243, 16 · 1, 232, 1458, 928, 25 … Σ 58, 368, 2644, 21496, 195680, 1973760, …
  (−∞, 0]nonnegative
• A075497 → reverse → D
  2 · 6, 8 · 12, 56, 24 · 20, 200, 360, 64 … Σ 0, 2, 14, 92, 644, 4854, 39290, 340472, …
  (−∞, 0]nonnegative
• A166341 → D
  10, 2 · 23, 46, 3 · 50, 276, 150, 4 … Σ 12, 72, 480, 3600, 30240, 282240, …
  (−∞, 0]nonnegative
• A062145 → D
  4 · 10, 20 · 18, 90, 60 · 28, 252, 420, 140 … Σ 0, 4, 30, 168, 840, 3960, 18018, 80080, …
  (−∞, 0]nonnegative
• A143495 → D+1 → x ↦ x/(x+1)
  1 · 4, 5 · 16, 41, 26 · 64, 253, 329, 141 … Σ 1, 9, 83, 787, 7697, 77865, 816627, …
  [−1, 0]nonnegative
• A330381: Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.
  1 · 1, 2 · 1, 5, 3 · 1, 9, 13, 4 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  (−∞, 0)nonnegative
• A053117 → x ↦ x+1
  1 · 2, 2 · 3, 8, 4 · 4, 20, 24, 8 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−2, 0)nonnegative
• A027466 → basis A013611 → θ+1
  1 · 8, 8 · 64, 128, 48 · 512, 1536, 1152, 256 … Σ 1, 16, 240, 3456, 48384, 663552, …
  (−∞, 0]nonnegativerational roots
• A132440 → D+1 → (1/n)D²+(2/n)D+1/n → reverse
  1 · 1, 3 · 1, 6, 6 · 1, 9, 18, 6 … Σ 1, 4, 13, 34, 73, 136, 229, 358, …
  (−∞, 0]nonnegative
• A118931 → basis A007318
  1 · 1 · 1 · 3, 2 … Σ 1, 1, 1, 5, 17, 41, 241, 1261, …
  (−∞, −1]nonnegative
• A008955 → D+1
  1 · 2, 1 · 6, 13, 4 · 15, 112, 157, 36 … Σ 1, 3, 23, 320, 7040, 225540, 9936180, …
  (−∞, 0]nonnegative
• A038328 → basis A013611 → basis A007318
  484, 352, 64 · 10648, 11616, 4224, 512 … Σ 900, 27000, 810000, 24300000, …
  (−∞, −1]nonnegativerational roots
• A122542 → θ+1 → reverse
  1 · 2 · 3, 4 · 4, 12, 4 … Σ 1, 2, 7, 20, 57, 158, 431, 1160, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) → x ↦ x+1 → D
  2 · 16 · 120, 32 · 960, 544 … Σ 0, 0, 2, 16, 152, 1504, 16608, 201216, …
  (−∞, −1]nonnegative
• A249247 → θ+1
  1 · 1, 2 · 2, 6 · 6, 20, 3 … Σ 1, 3, 8, 29, 129, 691, 4348, 31509, …
  (−∞, 0]nonnegative
• A118931 → basis A013610 → x ↦ x+1
  1 · 1 · 1 · 9, 6 … Σ 1, 1, 1, 15, 57, 141, 2241, 14211, …
  (−∞, −1]nonnegative
• A001263 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 5, 30, 41 · 14, 126, 354, 306 … Σ 1, 8, 76, 800, 8992, 105728, 1284544, …
  [−1, 0]nonnegative
• A038259 → basis A013611
  1 · 11, 20 · 121, 440, 400 · 1331, 7260, 13200, 8000 … Σ 1, 31, 961, 29791, 923521, 28629151, …
  [−1, 0]nonnegativerational roots
• A116666 → reverse → x ↦ x+1
  1 · 4, 1 · 12, 8, 1 · 32, 36, 12, 1 … Σ 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, −1]nonnegativerational roots
• A132393 → basis A013611 → D+1
  1 · 5, 4 · 14, 44, 16 · 50, 236, 288, 64 … Σ 1, 9, 74, 638, 5944, 60216, 662640, …
  (−∞, 0]nonnegative
• A104684 → basis A013610 → θ+1
  1 · 3, 6 · 13, 48, 27 · 63, 342, 405, 108 … Σ 1, 9, 88, 918, 9834, 106326, 1152664, …
  (−∞, 0]nonnegative
• A038322 → basis A013609 → reverse
  1 · 16, 19 · 256, 608, 361 · 4096, 14592, 17328, 6859 … Σ 1, 35, 1225, 42875, 1500625, 52521875, …
  (−∞, 0]nonnegativerational roots
• A063967 → basis A013611 → D
  4 · 20, 32 · 80, 256, 192 · 288, 1376, 2112, 1024 … Σ 0, 4, 52, 528, 4800, 41040, 337392, …
  (−∞, 0]nonnegative
• A249250 → x ↦ x+1 → D+1
  1 · 7, 3 · 22, 18, 3 · 76, 96, 33, 3 … Σ 1, 10, 43, 208, 949, 4294, 19135, …
  (−∞, 0]nonnegative
• A281260 → D
  2 · 2, 6 · 2, 16, 12 · 2, 30, 60, 20 … Σ 0, 2, 8, 30, 112, 420, 1584, 6006, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A008277 → D
  3, 2 · 7, 12, 3 · 15, 50, 30, 4 · 31, 180, 195, 60, 5 … Σ 5, 22, 99, 471, 2386, 12867, 73681, …
  (−∞, 0]nonnegative
• A109692 → basis A007318 → θ+1
  1 · 2, 2 · 8, 20, 9 · 48, 200, 204, 60 … Σ 1, 4, 37, 512, 9297, 207828, 5504229, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → (θ+1)² → reverse
  1 · -4 · 9 · -16 … Σ 1, -4, 9, -16, 25, -36, 49, -64, …
  (−∞, 0]rational roots
• A038322 → basis A013609 → D+1
  1 · 35, 16 · 969, 1120, 256 … Σ 1, 51, 2345, 101675, 4244625, …
  (−∞, 0]nonnegativerational roots
• A075504 → x ↦ x/(x+1)
  81, 189, 109 · 729, 2754, 3375, 1351 … Σ 379, 8209, 200737, 5690035, 186739723, …
  [−1, 0]nonnegative
• A103633 → Veronese 2 (j=1) → θ+1
  1 · 1 · 0, 2 · 0, 4 … Σ 0, 1, 1, 2, 4, 5, 11, 13, …
  (−∞, 0]nonnegative
• A116666 → D+1
  1 · 4, 3 · 7, 16, 5 · 10, 39, 36, 7 … Σ 1, 7, 28, 92, 272, 752, 1984, 5056, …
  (−∞, 0]nonnegative
• A136124 → basis A008277 → reverse
  1 · 1, 3 · 1, 8, 12 · 1, 15, 60, 60 … Σ 1, 4, 21, 136, 1045, 9276, 93289, …
  (−∞, 0]nonnegative
• A308737 → x ↦ x+1
  1 · 4, 3 · 16, 22, 7 · 64, 124, 76, 15 … Σ 1, 7, 45, 279, 1701, 10287, 61965, …
  (−∞, −1]nonnegativerational roots