Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A081579 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 8, 40, 41 · 24, 180, 414, 285 … Σ 1, 7, 89, 903, 9881, 105287, 1132249, …
  [−1, 0]nonnegative
• A113214 → θ+1 → D+1
  1 · 4, 2 · 6, 12, 3 · 4, 28, 24, 4 … Σ 1, 6, 21, 60, 150, 348, 763, 1608, …
  (−∞, 0]nonnegative
• A102547 → basis A007318 → x ↦ x+1
  1 · 1 · 1 · 3, 1 … Σ 1, 1, 1, 4, 7, 10, 22, 43, …
  (−∞, −1]nonnegative
• A161198 → basis A013611 → x ↦ x+1
  1 · 11, 8 · 143, 192, 64 · 2145, 4024, 2496, 512 … Σ 1, 19, 399, 9177, 229425, 6194475, …
  (−∞, −1]nonnegativerational roots
• A269945 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 7, 7 … Σ 1, 1, 3, 15, 121, 1345, 19371, 351663, …
  [−1, 0]nonnegative
• A131431: 3n + 1 preceded by n zeros.
  1 · 0, 4 · 0, 0, 7 · 0, 0, 0, 10 … Σ 1, 4, 7, 10, 13, 16, 19, 22, …
  [−1, 0]nonnegativerational roots
• A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
  0, 1 · 0, 3, 3 · 0, 6, 12, 6 · 0, 10, 30, 30, 10 … Σ 0, 0, 1, 6, 24, 80, 240, 672, …
  (−∞, 0]nonnegativerational roots
• A123519 → reverse
  4, 6, 1 · 8, 20, 12, 1 · 16, 56, 60, 20, 1 … Σ 11, 41, 153, 571, 2131, 7953, 29681, …
  (−∞, 0]nonnegative
• A038226 → basis A007318 → θ+1
  121, 352, 192 · 1331, 5808, 6336, 2048 … Σ 665, 15523, 349809, 7688939, 165898633, …
  (−∞, 0]nonnegativerational roots
• A038277 → x ↦ x/(x+1) → θ+1
  1 · 7, 36 · 49, 504, 972 · 343, 5292, 20412, 23328 … Σ 1, 43, 1525, 49375, 1515625, 44921875, …
  (−∞, 0]nonnegativerational roots
• A132393 → D → D+1
  1 · 3, 2 · 8, 12, 3 · 28, 58, 30, 4 … Σ 0, 1, 5, 23, 120, 724, 5012, 39332, …
  (−∞, 0]nonnegative
• A181371 → (θ+1)²
  1 · 3 · 8, 4 · 21, 24 … Σ 1, 3, 12, 45, 164, 585, 2052, 7101, …
  (−∞, 0]nonnegative
• A062264 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 6, 11 · 28, 98, 85 · 120, 612, 1032, 575 … Σ 1, 17, 211, 2339, 24598, 251510, …
  [−1, 0]nonnegative
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → θ+1
  1 · 4, 4 · 14, 28, 9 · 44, 132, 90, 16 … Σ 1, 8, 51, 282, 1413, 6588, 29079, …
  (−∞, 0]nonnegative
• A038323 → basis A013609 → D+1
  1 · 38, 18 · 1120, 1368, 324 … Σ 1, 56, 2812, 132848, 6035920, …
  (−∞, 0]nonnegativerational roots
• A013622 → θ+1
  1 · 3, 10 · 9, 60, 75 · 27, 270, 675, 500 … Σ 1, 13, 144, 1472, 14336, 135168, …
  (−∞, 0]nonnegativerational roots
• A391533 → (θ+1)²
  4, 16, 9 · 4, 32, 45, 16 · 4, 48, 117, 96, 25 … Σ 29, 97, 290, 808, 2144, 5488, 13664, …
  (−∞, 0]nonnegative
• A207824 → x ↦ x+1 → D+1
  1 · 7, 1 · 47, 14, 1 · 310, 142, 21, 1 … Σ 1, 8, 62, 474, 3585, 26876, 200003, …
  (−∞, 0]nonnegative
• A046716 → basis A013611 → θ+1
  1 · 2, 8 · 5, 40, 48 · 16, 200, 528, 256 … Σ 1, 10, 93, 1000, 14801, 315886, …
  (−∞, 0]nonnegative
• A062145 → D → (θ+1)²
  4 · 10, 80 · 18, 360, 540 · 28, 1008, 3780, 2240 … Σ 0, 4, 90, 918, 7056, 46800, 283338, …
  (−∞, 0]nonnegative
• A136011 → D
  1 · 3 · 7, 2 · 15, 12 … Σ 0, 0, 1, 3, 9, 27, 84, 273, …
  (−∞, 0]nonnegative
• A038255 → basis A013609 → D+1
  1 · 9, 2 · 77, 36, 4 · 637, 462, 108, 8 … Σ 1, 11, 117, 1215, 12393, 124659, …
  (−∞, 0]nonnegativerational roots
• A356546 → basis A013609 → θ+1
  1 · 4, 8 · 24, 96, 72 · 160, 960, 1440, 640 … Σ 1, 12, 192, 3200, 53760, 903168, …
  (−∞, 0]nonnegativerational roots
• A114687 → basis A013609 → θ+1
  1 · 3, 8 · 11, 56, 48 · 45, 336, 576, 256 … Σ 1, 11, 115, 1213, 13013, 141727, …
  (−∞, 0]nonnegative
• A272866 → basis A013609 → D
  10, 8 · 100, 280, 240, 64 … Σ 0, 18, 684, 19494, 493848, 11728890, …
  (−∞, 0]nonnegative
• A136124 → basis A013611 → D
  4 · 28, 32 · 188, 384, 192 · 1368, 3808, 3456, 1024 … Σ 0, 4, 60, 764, 9656, 126376, 1738272, …
  (−∞, 0]nonnegative
• A085478 → reverse → D
  1 · 3, 2 · 5, 12, 3 · 7, 30, 30, 4 … Σ 0, 1, 5, 20, 71, 235, 744, 2285, …
  (−∞, 0]nonnegative
• A119462 → basis A007318
  1 · 2 · 4, 2 · 8, 6 … Σ 1, 2, 6, 14, 34, 82, 198, 478, …
  (−∞, −1]nonnegative
• A386789 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 3, 9 · 0, 4, 28, 44 … Σ 1, 2, 12, 76, 500, 3372, 23128, 160568, …
  [−1, 0]nonnegative
• A062145 → reverse → D+1
  1 · 5, 1 · 20, 12, 1 · 65, 81, 21, 1 … Σ 1, 6, 33, 168, 810, 3762, 17017, 75504, …
  (−∞, 0]nonnegative
• A106246 → basis A007318 → θ+1
  1 · 3, 2 · 6, 12, 3 · 10, 36, 27, 4 … Σ 1, 5, 21, 77, 256, 792, 2320, 6512, …
  (−∞, 0]nonnegative
• A081578 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 7, 21, 15 · 20, 90, 132, 63 … Σ 1, 5, 43, 305, 2299, 16985, 126307, …
  [−1, 0]nonnegative
• A274498 → D → D
  72 · 216 · 648, 1296 · 1944, 3888 … Σ 0, 0, 0, 0, 72, 216, 1944, 5832, …
  (−∞, 0]nonnegativerational roots
• A013610 → basis A013611 → D
  12 · 96, 288 · 576, 3456, 5184 … Σ 0, 12, 384, 9216, 196608, 3932160, …
  (−∞, 0]nonnegativerational roots
• A144697 → D
  3 · 10, 18 · 25, 134, 81 · 56, 652, 1128, 324 … Σ 0, 3, 28, 240, 2160, 21000, 221760, …
  (−∞, 0]nonnegative
• A127648 → (θ+1)² → D+1
  1 · 8, 8 · 0, 54, 27 · 0, 0, 192, 64 … Σ 0, 1, 16, 81, 256, 625, 1296, 2401, …
  (−∞, 0]nonnegativerational roots
• A091042 → basis A013610 → reverse
  1 · 9, 4 · 45, 60, 16 · 189, 504, 336, 64 … Σ 1, 13, 121, 1093, 9841, 88573, 797161, …
  (−∞, 0]nonnegative
• A038335 → D
  9 · 216, 162 · 3888, 5832, 2187 … Σ 0, 9, 378, 11907, 333396, 8751645, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x…
  1 · 0, 0, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 0, 1 … Σ 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0][0, ∞)nonnegative+1
• A091044 → basis A013611 → reverse
  1 · 8, 4 · 48, 64, 16 · 256, 640, 384, 64 … Σ 1, 12, 128, 1344, 14080, 147456, …
  (−∞, 0]nonnegative
• A120910 → basis A013609 → (θ+1)²
  3 · 9, 24 · 27, 144, 108 · 81, 648, 972, 384 … Σ 3, 33, 279, 2085, 14475, 95625, 609375, …
  (−∞, 0]nonnegative
• A272867 → x ↦ x+1 → (θ+1)²
  1 · 6, 24, 9 · 36, 288, 432, 192, 25 … Σ 1, 39, 973, 20371, 386841, 6896383, …
  (−∞, 0]nonnegative
• A085880 → basis A013610
  1 · 2, 3 · 8, 24, 18 · 40, 180, 270, 135 … Σ 1, 5, 50, 625, 8750, 131250, 2062500, …
  [−1, 0]nonnegativerational roots
• A114655 → θ+1 → x ↦ x/(x+1)
  2 · 4, 8 · 8, 40, 38 · 16, 144, 312, 192 … Σ 2, 12, 86, 664, 5322, 43620, 362910, …
  [−1, 0]nonnegative
• A272866 → basis A013610 → θ+1
  1 · 5, 30, 27 · 25, 300, 945, 1080, 405 … Σ 1, 62, 2755, 107648, 3926629, …
  (−∞, 0]nonnegative
• A345013 → x ↦ x/(x+1)
  1 · 4, 7 · 15, 50, 41 · 56, 273, 438, 231 … Σ 1, 11, 106, 998, 9367, 88085, 830980, …
  [−1, 0]nonnegative
• A169803 → basis A013611 → θ+1
  1 · 2, 8 · 3, 16 · 5, 40, 48 … Σ 1, 10, 19, 93, 232, 861, 2405, 7914, …
  (−∞, 0]nonnegative
• chebyshev_t → x ↦ x+1 → reverse
  1 · 1, 1 · 2, 4, 1 · 4, 12, 9, 1 … Σ 1, 2, 7, 26, 97, 362, 1351, 5042, …
  (−∞, 0]nonnegative
• A085880 → basis A013609 → (θ+1)²
  1 · 2, 8 · 8, 64, 72 · 40, 480, 1080, 640 … Σ 1, 10, 144, 2240, 35840, 580608, …
  (−∞, 0]nonnegative
• A038333 → basis A013610 → D+1
  1159, 1680, 441 · 29602, 73017, 52920, 9261 … Σ 3280, 164800, 7936000, 371200000, …
  (−∞, 0]nonnegativerational roots