Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A136572 → (θ+1)²
  1 · 0, 4 · 0, 0, 18 · 0, 0, 0, 96 … Σ 1, 4, 18, 96, 600, 4320, 35280, 322560, …
  [−1, 0]nonnegativerational roots
• narayana → D → D+1
  1 · 5, 2 · 18, 18, 3 · 50, 100, 42, 4 … Σ 0, 1, 7, 39, 196, 930, 4257, 19019, …
  (−∞, 0]nonnegative
• A011973 → D
  1 · 2 · 3, 2 · 4, 6 … Σ 0, 0, 1, 2, 5, 10, 20, 38, …
  (−∞, 0]nonnegative
• A274496 → x ↦ x+1
  1 · 2 · 4, 2 · 8, 4 … Σ 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A113214, shifted → Veronese 2 (j=1) → D+1
  -1 · 2 · -4, -1 · 8, 4 … Σ 0, -1, 2, -5, 12, -28, 64, -144, …
  (−∞, 0]
• A271703 → reverse
  1 · 1 · 1, 2 · 1, 6, 6 … Σ 1, 1, 3, 13, 73, 501, 4051, 37633, …
  (−∞, 0]nonnegative
• A166962 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 14, 14 · 1, 106, 278, 174 … Σ 1, 3, 29, 559, 16001, 652251, 35570181, …
  [−1, 0]nonnegative
• basis-map coefficients of A113214, shifted → basis A013609 → x ↦ x+1
  1 · -4, -2 · 16, 16, 4 · -64, -96, -48, -8 … Σ 1, -6, 36, -216, 1296, -7776, 46656, …
  (−∞, −1]rational roots
• A038338 → D+1
  432, 576, 144 · 6912, 15552, 10368, 1728 … Σ 1152, 34560, 995328, 27869184, …
  (−∞, 0]nonnegativerational roots
• A038333 → basis A013611 → D+1
  1425, 2632, 784 · 37183, 119700, 110544, 21952 … Σ 4841, 289379, 16507857, 912500347, …
  (−∞, 0]nonnegativerational roots
• A088996 → basis A013610 → (θ+1)²
  1 · 1, 12 · 3, 60, 162 · 15, 408, 2025, 2592 … Σ 1, 13, 225, 5040, 138816, 4535604, …
  (−∞, 0]nonnegative
• A103406 → Veronese 2 (j=1) → θ+1
  2 · 4 · 6, 8 · 8, 32 … Σ 0, 2, 4, 14, 40, 108, 280, 704, …
  (−∞, 0]nonnegative
• A371080 → x ↦ x+1
  1 · 1, 1 · 5, 6, 1 · 41, 55, 15, 1 … Σ 1, 2, 12, 112, 1408, 22192, 419904, …
  (−∞, −1]nonnegative
• A038319 → basis A013609 → θ+1
  256, 640, 300 · 4096, 15360, 14400, 4000 … Σ 1196, 37856, 1160016, 34730176, …
  (−∞, 0]nonnegativerational roots
• A028338 → x ↦ x+1 → basis A007318
  1 · 3, 1 · 15, 8, 1 · 105, 71, 15, 1 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, −1]nonnegativerational roots
• A052553 → x ↦ x+1 → D+1
  1 · 1 · 3, 1 · 5, 2 … Σ 1, 1, 4, 7, 18, 37, 84, 179, …
  (−∞, 0]nonnegative
• A145596 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 14, 14 · 4, 32, 72, 48 … Σ 1, 6, 31, 156, 785, 3978, 20335, …
  [−1, 0]nonnegative
• A038305 → basis A013611 → D+1
  481, 600, 144 · 8281, 17316, 10800, 1728 … Σ 1225, 38125, 1140625, 33203125, …
  (−∞, 0]nonnegativerational roots
• A035607 → basis A013611 → D+1
  1 · 1 · 11, 8 · 39, 96, 32 … Σ 1, 1, 19, 167, 1425, 11593, 91571, …
  (−∞, 0]nonnegative
• A367270 → x ↦ x+1 → D+1
  1 · 1 · 5, 2 · 22, 18, 3 … Σ 1, 1, 7, 43, 261, 1581, 9563, 57751, …
  (−∞, 0]nonnegative
• A142075 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 10, 13 · 1, 25, 91, 75 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  [−1, 0]nonnegative
• A110441 → θ+1
  1 · 3, 2 · 7, 12, 3 · 15, 46, 27, 4 … Σ 1, 5, 22, 92, 372, 1468, 5688, 21728, …
  (−∞, 0]nonnegative
• A099040 → basis A013609 → θ+1
  1 · 2, 8 · 6, 40, 48 · 16, 160, 384, 256 … Σ 1, 10, 94, 816, 6780, 54648, 430920, …
  (−∞, 0]nonnegative
• A049374 → reverse
  1 · 1, 6 · 1, 18, 42 · 1, 36, 276, 336 … Σ 1, 7, 61, 649, 8245, 122215, 2069425, …
  (−∞, 0]nonnegative
• A028326 → basis A007318 → D
  2 · 8, 4 · 24, 24, 6 · 64, 96, 48, 8 … Σ 0, 2, 12, 54, 216, 810, 2916, 10206, …
  (−∞, −1]nonnegativerational roots
• A272867: Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  1 · 1, 4, 1 · 1, 8, 18, 8, 1 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  (−∞, 0]nonnegative
• A063967 → basis A013610 → D+1
  1 · 5, 3 · 21, 33, 9 · 76, 204, 153, 27 … Σ 1, 8, 63, 460, 3230, 22050, 147475, …
  (−∞, 0]nonnegative
• A200536 → basis A013610 → x ↦ x+1
  1 · 45, 57, 18 · 2025, 5130, 4869, 2052, 324 … Σ 1, 120, 14400, 1728000, 207360000, …
  (−∞, −1]nonnegativerational roots
• A122076 → (θ+1)²
  2 · 3, 8 · 7, 32, 18 · 18, 120, 135, 32 … Σ 2, 11, 57, 305, 1617, 8345, 41897, …
  (−∞, 0]nonnegative
• A038326 → x ↦ x+1 → D
  12 · 552, 288 · 19044, 19872, 5184 … Σ 0, 12, 840, 44100, 2058000, 90037500, …
  (−∞, 0]nonnegativerational roots
• A038329 → x ↦ x+1 → x ↦ x/(x+1)
  225, 540, 324 · 3375, 12150, 14580, 5832 … Σ 1089, 35937, 1185921, 39135393, …
  [−1, 0]nonnegativerational roots
• A225471 → basis A013609 → basis A007318
  1 · 6, 2 · 60, 32, 4 · 840, 568, 120, 8 … Σ 1, 8, 96, 1536, 30720, 737280, …
  (−∞, −1]nonnegativerational roots
• A227450 → reverse → θ+1
  2 · 4, 4 · 8, 24, 6 · 16, 96, 84, 8 … Σ 2, 8, 38, 204, 1202, 7656, 52222, …
  (−∞, 0]nonnegative
• A008459 → basis A013611 → θ+1
  1 · 2, 8 · 6, 48, 48 · 20, 240, 576, 256 … Σ 1, 10, 102, 1092, 11910, 130476, …
  (−∞, 0]nonnegative
• A123160 → reverse
  1 · 1, 1 · 3, 4, 1 · 10, 18, 9, 1 … Σ 1, 2, 8, 38, 192, 1002, 5336, 28814, …
  (−∞, 0]nonnegative
• A056939 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 6, 6 · 1, 13, 33, 22 … Σ 1, 3, 13, 69, 417, 2763, 19609, 146793, …
  (−∞, 0]nonnegative
• A008299 → θ+1
  1 · 1 · 1, 6 · 1, 20 … Σ 1, 1, 7, 21, 96, 428, 2129, 11287, …
  (−∞, 0]nonnegative
• A257610 → θ+1
  1 · 2, 4 · 4, 40, 12 · 8, 264, 396, 32 … Σ 1, 6, 56, 700, 10920, 203840, 4426240, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → Veronese 2 → x ↦ x+1
  1 · 2, 2 · 24, 48, 24 · 720, 2160, 2160, 720 … Σ 1, 0, 4, 0, 96, 0, 5760, 0, …
  (−∞, −1]nonnegativerational roots
• narayana → basis A013611 → D
  4 · 20, 32 · 84, 288, 192 · 336, 1792, 2688, 1024 … Σ 0, 4, 52, 564, 5840, 59560, 604284, …
  (−∞, 0]nonnegative
• A094638 → basis A013609 → reverse
  1 · 2, 2 · 8, 14, 6 · 48, 116, 92, 24 … Σ 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0]nonnegativerational roots
• A225473 → θ+1
  1 · 3, 8 · 9, 80, 96 · 27, 632, 2016, 1536 … Σ 1, 11, 185, 4211, 120497, 4148891, …
  (−∞, 0]nonnegative
• A136572 → basis A013609 → D+1
  1 · 3, 2 · 10, 24, 8 · 42, 180, 216, 48 … Σ 1, 5, 42, 486, 7128, 126360, 2624400, …
  (−∞, 0]nonnegativerational roots
• A038319 → reverse
  25, 110, 121 · 125, 825, 1815, 1331 … Σ 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0]nonnegativerational roots
• A105278 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 24, 108, 156, 73 … Σ 1, 5, 37, 361, 4361, 62701, 1044205, …
  [−1, 0]nonnegative
• A085880 → D → (θ+1)²
  1 · 4, 16 · 15, 120, 135 · 56, 672, 1512, 896 … Σ 0, 1, 20, 270, 3136, 33600, 342144, …
  (−∞, 0]nonnegative
• A027467 → basis A013610 → D
  3 · 96, 18 · 2304, 864, 81 · 49152, 27648, 5184, 324 … Σ 0, 3, 114, 3249, 82308, 1954815, …
  (−∞, 0]nonnegativerational roots
• A147716 → basis A013611 → basis A013610
  1 · 19, 12 · 361, 456, 144 · 6859, 12996, 8208, 1728 … Σ 1, 31, 961, 29791, 923521, 28629151, …
  (−∞, −1]nonnegativerational roots
• A091044 → basis A013609 → D+1
  1 · 8, 4 · 48, 56, 12 · 256, 512, 256, 32 … Σ 1, 12, 116, 1056, 9296, 79936, 675392, …
  (−∞, 0]nonnegative
• A090452 → θ+1
  1 · 1, 6, 6 · 1, 14, 48, 60, 25 … Σ 1, 13, 148, 1644, 18192, 201616, …
  (−∞, 0]nonnegative