Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A290319 → x ↦ x+1 → D
  1 · 8, 2 · 92, 36, 3 · 1408, 688, 96, 4 … Σ 0, 1, 10, 131, 2196, 45189, 1105182, …
  (−∞, 0]nonnegative
• A091042 → basis A007318 → basis A007318
  1 · 7, 3 · 41, 30, 5 · 239, 245, 77, 7 … Σ 1, 10, 76, 568, 4240, 31648, 236224, …
  (−∞, −1]nonnegative
• A103327 → reverse
  1 · 1, 3 · 1, 10, 5 · 1, 21, 35, 7 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0]nonnegative
• A088996 → basis A013610 → basis A007318
  1 · 4, 3 · 36, 51, 18 · 504, 1038, 711, 162 … Σ 1, 7, 105, 2415, 74865, 2919735, …
  (−∞, −1]nonnegativerational roots
• A122848 → basis A013610 → basis A007318
  1 · 4, 3 · 20, 27, 9 · 112, 216, 135, 27 … Σ 1, 7, 56, 490, 4606, 45962, 482944, …
  (−∞, −1]nonnegative
• basis-map coefficients of A297191, shifted → basis A013611
  1 · 1, 8, 16 · 1, 16, 96, 256, 256 … Σ 1, 25, 625, 15625, 390625, 9765625, …
  [−1, 0]nonnegativerational roots
• A038325 → basis A007318 → basis A007318
  1 · 33, 11 · 1089, 726, 121 … Σ 1, 44, 1936, 85184, 3748096, 164916224, …
  (−∞, −1]nonnegativerational roots
• A397048 → D+1
  1 · 2, 1 · 9, 7, 1 · 64, 61, 15, 1 … Σ 1, 3, 17, 141, 1540, 20878, 338233, …
  (−∞, 0]nonnegative
• A136572 → basis A013611 → (θ+1)²
  1 · 1, 16 · 2, 64, 288 · 6, 288, 2592, 6144 … Σ 1, 17, 354, 9030, 274200, 9675000, …
  (−∞, 0]nonnegative
• A156992 → basis A013611 → θ+1
  1 · 4, 16 · 24, 192, 288 · 192, 2304, 6912, 6144 … Σ 1, 20, 504, 15552, 570240, 24261120, …
  (−∞, 0]nonnegativerational roots
• A200536 → basis A013609 → reverse
  1 · 8, 14, 6 · 64, 224, 292, 168, 36 … Σ 1, 28, 784, 21952, 614656, 17210368, …
  (−∞, 0]nonnegativerational roots
• A152842 → reverse → x ↦ x+1
  1 · 4, 1 · 8, 6, 1 · 32, 32, 10, 1 … Σ 1, 5, 15, 75, 225, 1125, 3375, 16875, …
  (−∞, −1]nonnegativerational roots
• A062145 → x ↦ x+1 → (θ+1)²
  1 · 5, 16 · 21, 120, 90 · 84, 672, 945, 320 … Σ 1, 21, 231, 2021, 15849, 116793, …
  (−∞, 0]nonnegative
• A178820 → basis A007318 → basis A007318
  1 · 12, 4 · 90, 60, 10 · 540, 540, 180, 20 … Σ 1, 16, 160, 1280, 8960, 57344, 344064, …
  (−∞, −1]nonnegativerational roots
• A114656 → basis A007318 → D+1
  1 · 4, 1 · 19, 10, 1 · 96, 81, 18, 1 … Σ 1, 5, 30, 196, 1344, 9504, 68640, …
  (−∞, 0]nonnegative
• A038221 → basis A013611
  1 · 6, 12 · 36, 144, 144 · 216, 1296, 2592, 1728 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  [−1, 0]nonnegativerational roots
• A038335 → x ↦ x/(x+1)
  1 · 12, 21 · 144, 504, 441 · 1728, 9072, 15876, 9261 … Σ 1, 33, 1089, 35937, 1185921, 39135393, …
  [−1, 0]nonnegativerational roots
• A059344 → basis A007318 → D
  2 · 6 · 36, 24 · 140, 120 … Σ 0, 0, 2, 6, 60, 260, 2190, 11802, …
  (−∞, 0]nonnegative
• A027467: Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).
  1 · 15, 1 · 225, 30, 1 · 3375, 675, 45, 1 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, −1]nonnegativerational roots
• A204420 → basis A007318 → θ+1
  1 · 1, 2 · 9, 24, 9 · 225, 690, 405, 60 … Σ 1, 3, 42, 1380, 82320, 7771680, …
  (−∞, 0]nonnegative
• A113953 → basis A013610 → x ↦ x/(x+1)
  1 · 1, 4 · 3, 18, 24 · 5, 48, 144, 128 … Σ 1, 5, 45, 325, 2525, 19125, 146125, …
  [−1, 0]nonnegative
• A038328 → basis A013609 → D
  112, 32 · 2352, 1344, 192 · 43904, 37632, 10752, 1024 … Σ 144, 3888, 93312, 2099520, 45349632, …
  (−∞, 0]nonnegativerational roots
• narayana → basis A013610 → reverse
  1 · 3, 2 · 9, 15, 5 · 27, 81, 63, 14 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, 0]nonnegative
• A034870 → basis A013610 → basis A007318
  1 · 25, 30, 9 · 625, 1500, 1350, 540, 81 … Σ 1, 64, 4096, 262144, 16777216, …
  (−∞, −1]nonnegativerational roots
• A008292: Eulerian
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0)nonnegative
• A091042 → reverse
  1 · 3, 1 · 5, 10, 1 · 7, 35, 21, 1 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0]nonnegative
• A169803 → basis A013609 → D
  2 · 4 · 10, 8 · 20, 24 … Σ 0, 2, 4, 18, 44, 136, 348, 950, …
  (−∞, 0]nonnegative
• A124932 → basis A007318 → x ↦ x+1
  1 · 8, 3 · 45, 33, 6 · 216, 234, 84, 10 … Σ 1, 11, 84, 544, 3200, 17664, 93184, …
  (−∞, −1]nonnegativerational roots
• A201637 → reverse (shift +1) → D
  1 · 2, 2 · 6, 16, 3 · 24, 116, 66, 4 … Σ 0, 1, 4, 25, 210, 2205, 27720, 405405, …
  (−∞, 0]nonnegative
• A038243 → basis A013609
  1 · 6, 2 · 36, 24, 4 · 216, 216, 72, 8 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, −1]nonnegativerational roots
• A124932 → basis A007318
  1 · 5, 3 · 18, 21, 6 · 56, 96, 54, 10 … Σ 1, 8, 45, 216, 945, 3888, 15309, 58320, …
  (−∞, −1]nonnegativerational roots
• A126216 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 14, 63, 93, 45 … Σ 1, 5, 31, 215, 1597, 12425, 99955, …
  [−1, 0]nonnegative
• A033282 → x ↦ x+1 → θ+1
  1 · 3, 4 · 11, 30, 15 · 45, 186, 189, 56 … Σ 1, 7, 56, 476, 4187, 37645, 343630, …
  (−∞, 0]nonnegative
• A062145 → x ↦ x+1 → x ↦ x+1
  1 · 9, 4 · 61, 50, 10 · 377, 438, 165, 20 … Σ 1, 13, 121, 1000, 7834, 59719, 448417, …
  (−∞, −1]nonnegative
• A038277 → θ+1
  1 · 7, 22 · 49, 308, 363 · 343, 3234, 7623, 5324 … Σ 1, 29, 720, 16524, 361584, 7663248, …
  (−∞, 0]nonnegativerational roots
• A161198 → basis A007318 → (θ+1)²
  1 · 3, 8 · 15, 64, 36 · 105, 568, 540, 128 … Σ 1, 11, 115, 1341, 17785, 267099, …
  (−∞, 0]nonnegative
• A027467 → reverse
  1 · 1, 15 · 1, 30, 225 · 1, 45, 675, 3375 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, 0]nonnegativerational roots
• A038231 → Veronese 2 (j=1) → x ↦ x+1
  1 · 8 · 49, 1 · 272, 16 … Σ 0, 1, 8, 50, 288, 1604, 8800, 47944, …
  (−∞, −1]nonnegative
• A219512 → D
  3 · 12, 30 · 33, 282, 315 · 78, 1572, 5490, 3780 … Σ 0, 3, 42, 630, 10920, 218400, 4979520, …
  (−∞, 0]nonnegative
• A008459 → basis A013610 → θ+1
  1 · 2, 6 · 6, 36, 27 · 20, 180, 324, 108 … Σ 1, 8, 69, 632, 5905, 55440, 520485, …
  (−∞, 0]nonnegative
• A200536 → basis A013610 → (θ+1)²
  1 · 6, 84, 162 · 36, 1008, 5913, 12096, 8100 … Σ 1, 252, 27153, 2225880, 158502825, …
  (−∞, 0]nonnegative
• A347056 → θ+1
  1 · 3, 8 · 6, 32, 30 · 10, 80, 150, 80 … Σ 1, 11, 68, 320, 1280, 4592, 15232, …
  (−∞, 0]nonnegative
• A135065 → x ↦ x+1 → basis A007318
  1 · 12, 4 · 81, 54, 9 · 432, 432, 144, 16 … Σ 1, 16, 144, 1024, 6400, 36864, 200704, …
  (−∞, −1]nonnegativerational roots
• A038208 → Veronese 2 → D
  4 · 24 · 96, 32 · 320, 320 … Σ 0, 0, 4, 24, 128, 640, 3072, 14336, …
  (−∞, 0]nonnegative
• A038333 → basis A013611 → D
  1064, 1568 · 30324, 89376, 65856 … Σ 2632, 185556, 11628176, 683155340, …
  (−∞, 0]nonnegativerational roots
• A038319 → x ↦ x+1 → D+1
  416, 210, 25 · 7936, 6240, 1575, 125 … Σ 651, 15876, 379701, 8946126, 208289151, …
  (−∞, 0]nonnegativerational roots
• A099093 → x ↦ x−2
  1 · -6, 3 · 30, -33, 9 · -144, 252, -144, 27 … Σ 1, -3, 6, -9, 9, 0, -27, 81, …
  (2/3, 2]
• A156992 → basis A013610 → (θ+1)²
  1 · 4, 24 · 24, 288, 486 · 192, 3456, 11664, 10368 … Σ 1, 28, 798, 25680, 939000, 38700000, …
  (−∞, 0]nonnegative
• A034870 → Veronese 2 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 8, 8 · 1, 18, 48, 32 … Σ 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  [−1, 0]nonnegative
• A078812 → (θ+1)²
  1 · 2, 4 · 3, 16, 9 · 4, 40, 54, 16 … Σ 1, 6, 28, 114, 427, 1512, 5141, 16950, …
  (−∞, 0]nonnegative