Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038243 → basis A013611 → D
  4 · 48, 32 · 432, 576, 192 · 3456, 6912, 4608, 1024 … Σ 0, 4, 80, 1200, 16000, 200000, 2400000, …
  (−∞, 0]nonnegativerational roots
• A274498 → D → (θ+1)²
  6 · 18 · 36, 288 · 108, 864 … Σ 0, 0, 6, 18, 324, 972, 8586, 25758, …
  (−∞, 0]nonnegative
• A013620 → Veronese 2 → D
  9 · 54 · 216, 162 · 720, 1620 … Σ 0, 0, 9, 54, 378, 2340, 14067, 82026, …
  (−∞, 0]nonnegative
• A027466 → (θ+1)²
  1 · 7, 4 · 49, 56, 9 · 343, 588, 189, 16 … Σ 1, 11, 114, 1136, 11008, 104448, …
  (−∞, 0]nonnegative
• A038326 → x ↦ x+1 → reverse
  1 · 12, 23 · 144, 552, 529 · 1728, 9936, 19044, 12167 … Σ 1, 35, 1225, 42875, 1500625, 52521875, …
  (−∞, 0]nonnegativerational roots
• A103633 → basis A007318 → reverse
  1 · 1, 1 · 1, 3, 2 · 1, 4, 5, 2 … Σ 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, 0]nonnegativerational roots
• A038226 → x ↦ x/(x+1)
  9, 66, 121 · 27, 297, 1089, 1331 … Σ 196, 2744, 38416, 537824, 7529536, …
  [−1, 0]nonnegativerational roots
• A013612 → x ↦ x+1
  36, 60, 25 · 216, 540, 450, 125 … Σ 121, 1331, 14641, 161051, 1771561, …
  (−∞, −1]nonnegativerational roots
• A008517 → D+1
  1 · 3, 2 · 9, 20, 6 · 23, 138, 130, 24 … Σ 1, 5, 35, 315, 3465, 45045, 675675, …
  (−∞, 0]nonnegative
• A008277 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 15, 22, 9, 1 … Σ 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  (−∞, −1]nonnegative
• A113953 → x ↦ x−4
  1 · -4, 1 · 8, -6, 1 · 0, 16, -8, 1 … Σ 1, -3, 3, 9, -45, 81, 27, -567, …
  (−4, 4]
• chebyshev_t → x ↦ x+1 → D
  1 · 4, 4 · 9, 24, 12 · 16, 80, 96, 32 … Σ 0, 1, 8, 45, 224, 1045, 4680, 20377, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → θ+1 → x ↦ x+1
  1 · -2, -2 · 3, 6, 3 · -4, -12, -12, -4 … Σ 1, -4, 12, -32, 80, -192, 448, -1024, …
  (−∞, −1]rational roots
• A013618 → basis A013610 → reverse
  1089, 792, 144 · 35937, 39204, 14256, 1728 … Σ 2025, 91125, 4100625, 184528125, …
  (−∞, 0]nonnegativerational roots
• A142075 → D
  2 · 8, 8 · 22, 88, 24 · 52, 528, 624, 64 … Σ 0, 2, 16, 134, 1268, 13586, 163560, …
  (−∞, 0]nonnegative
• A120909 → x ↦ x/(x+1)
  3 · 3, 9 · 3, 18, 27 · 3, 27, 81, 81 … Σ 3, 12, 48, 192, 768, 3072, 12288, …
  [−1, 0]nonnegativerational roots
• A132166 → D
  42, 2 · 1113, 126, 3 · 23184, 4662, 252, 4 … Σ 44, 1242, 28102, 555077, 10012512, …
  (−∞, 0]nonnegative
• A122848 → basis A013609 → x ↦ x/(x+1)
  1 · 1, 3 · 2, 10, 12 · 4, 30, 72, 54 … Σ 1, 4, 24, 160, 1216, 9984, 88576, …
  [−1, 0]nonnegative
• A100862 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 14, 63, 93, 45 … Σ 1, 5, 31, 215, 1633, 13325, 115615, …
  [−1, 0]nonnegative
• A013612 → basis A013609 → (θ+1)²
  36, 480, 900 · 216, 4320, 16200, 16000 … Σ 1416, 36736, 864256, 19070976, …
  (−∞, 0]nonnegative
• A027467 → (θ+1)²
  1 · 15, 4 · 225, 120, 9 · 3375, 2700, 405, 16 … Σ 1, 19, 354, 6496, 117760, 2113536, …
  (−∞, 0]nonnegative
• A099097 → reverse
  1 · 3 · 9, 1 · 27, 6 … Σ 1, 3, 10, 33, 109, 360, 1189, 3927, …
  (−∞, 0]nonnegative
• A272866 → Veronese 2 → θ+1
  1 · 1, 2 · 1, 22, 3 · 1, 60, 90, 4 … Σ 1, 3, 26, 155, 939, 5467, 31252, …
  (−∞, 0]nonnegative
• A131431 → D+1 → x ↦ x/(x+1)
  1 · 4, 8 · 0, 14, 21 · 0, 0, 30, 40 … Σ 1, 12, 35, 70, 117, 176, 247, 330, …
  [−1, 0]nonnegativerational roots
• A038220 → basis A013611 → θ+1
  1 · 5, 16 · 25, 160, 192 · 125, 1200, 2880, 2048 … Σ 1, 21, 377, 6253, 98865, 1513733, …
  (−∞, 0]nonnegativerational roots
• A038301 → basis A013609 → x ↦ x/(x+1)
  400, 1680, 1764 · 8000, 50400, 105840, 74088 … Σ 3844, 238328, 14776336, 916132832, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A030528, shifted → basis A013610 → D
  -3 · 6, 18 · -9, -54, -81 · 12, 108, 324, 324 … Σ 0, -3, 24, -144, 768, -3840, 18432, …
  (−∞, 0]rational roots
• A038331 → basis A013609 → reverse
  100, 340, 289 · 1000, 5100, 8670, 4913 … Σ 729, 19683, 531441, 14348907, …
  (−∞, 0]nonnegativerational roots
• A088996 → basis A013609 → x ↦ x+1
  1 · 3, 2 · 21, 26, 8 · 231, 412, 244, 48 … Σ 1, 5, 55, 935, 21505, 623645, 21827575, …
  (−∞, −1]nonnegativerational roots
• A232535 → basis A013610 → D+1
  1 · 9, 6 · 54, 96, 27 · 288, 906, 657, 108 … Σ 1, 15, 177, 1959, 20913, 217743, …
  (−∞, 0]nonnegative
• A299989 → reverse
  1 · 1, 4, 3 · 1, 8, 22, 24, 9 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, 0]nonnegativerational roots
• A142460 → D+1
  1 · 2, 1 · 13, 14, 1 · 84, 249, 86, 1 … Σ 1, 3, 28, 420, 8568, 219912, 6785856, …
  (−∞, 0]nonnegative
• A038291 → x ↦ x/(x+1)
  1 · 9, 10 · 81, 180, 100 · 729, 2430, 2700, 1000 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  [−1, 0]nonnegativerational roots
• A121314 → D+1
  1 · 1, 1 · 1, 3, 1 · 1, 7, 6, 1 … Σ 1, 2, 5, 15, 46, 139, 413, 1210, …
  (−∞, 0]nonnegative
• A327997 → x ↦ x+1 → basis A007318
  1 · 5, 1 · 27, 11, 1 · 159, 98, 18, 1 … Σ 1, 6, 39, 276, 2133, 18018, 166203, …
  (−∞, −1]nonnegative
• A027467 → basis A013609 → x ↦ x+1
  1 · 18, 2 · 324, 72, 4 · 5832, 1944, 216, 8 … Σ 1, 20, 400, 8000, 160000, 3200000, …
  (−∞, −1]nonnegativerational roots
• A133080 → basis A013609 → (θ+1)²
  1 · 2, 8 · 1, 16, 36 · 2, 40, 144, 128 … Σ 1, 10, 53, 314, 1161, 5994, 19197, …
  (−∞, 0]nonnegative
• A130850: Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.
  1 · 1, 1 · 2, 3, 1 · 6, 12, 7, 1 … Σ 1, 2, 6, 26, 150, 1082, 9366, 94586, …
  (−∞, −1]nonnegative
• A038322 → x ↦ x/(x+1)
  1 · 11, 19 · 121, 418, 361 · 1331, 6897, 11913, 6859 … Σ 1, 30, 900, 27000, 810000, 24300000, …
  [−1, 0]nonnegativerational roots
• A109983 → D+1
  1 · 1, 5, 2 · 0, 2, 19, 30, 6 … Σ 1, 8, 57, 384, 2505, 16008, 100849, …
  (−∞, 0]nonnegative
• A207606 → reverse → x ↦ x+1
  13, 15, 4 · 34, 58, 31, 5 · 89, 201, 162, 54, 6 … Σ 32, 128, 512, 2048, 8192, 32768, …
  (−∞, −1]nonnegative
• A151775 → x ↦ x+1
  1 · 3, 2 · 57, 76, 24 · 2763, 5462, 3480, 720 … Σ 1, 5, 157, 12425, 1836697, 436366445, …
  (−∞, −1]nonnegative
• A105070 → θ+1
  1 · 2 · 3, 4 · 4, 16 … Σ 1, 2, 7, 20, 57, 158, 431, 1160, …
  (−∞, 0]nonnegative
• A099040 → basis A007318 → reverse
  1 · 2, 2 · 4, 10, 6 · 8, 32, 40, 16 … Σ 1, 4, 20, 96, 464, 2240, 10816, 52224, …
  (−∞, 0]nonnegative
• A056241 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 18, 4 … Σ 1, 3, 10, 35, 123, 427, 1460, 4923, …
  (−∞, 0]nonnegative
• A027465 → θ+1
  1 · 3, 2 · 9, 12, 3 · 27, 54, 27, 4 … Σ 1, 5, 24, 112, 512, 2304, 10240, 45056, …
  (−∞, 0]nonnegativerational roots
• A136124 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 5 · 12, 38, 30 · 60, 274, 416, 210 … Σ 1, 8, 80, 960, 13440, 215040, 3870720, …
  [−1, 0]nonnegativerational roots
• A113214 → basis A013610 → (θ+1)²
  1 · 3, 12 · 4, 60, 81 · 7, 156, 567, 432 … Σ 1, 15, 145, 1162, 8378, 56412, 361976, …
  (−∞, 0]nonnegative
• A099097 → D
  3 · 1, 18 · 0, 12, 81 · 0, 2, 81, 324 … Σ 0, 3, 19, 93, 407, 1674, 6618, 25455, …
  (−∞, 0]nonnegative
• A094816 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 32, 54, 16 … Σ 1, 5, 22, 103, 543, 3273, 22512, …
  (−∞, 0]nonnegative