Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A286718 → basis A013609 → D+1
  1 · 4, 2 · 24, 22, 4 · 212, 252, 84, 8 … Σ 1, 6, 50, 556, 7788, 131888, 2622352, …
  (−∞, 0]nonnegative
• A143685 → basis A007318
  1 · 2, 1 · 13, 13, 1 · 44, 66, 24, 1 … Σ 1, 3, 27, 135, 891, 5103, 31347, …
  (−∞, −1]nonnegative
• A188440 → basis A007318 → x ↦ x+1
  1 · 1 · 5, 2 · 5, 2 … Σ 1, 1, 7, 7, 49, 49, 343, 343, …
  (−∞, −1]nonnegativerational roots
• A274496 → basis A013610 → θ+1
  1 · 2 · 4, 12 · 8, 24 … Σ 1, 2, 16, 32, 220, 440, 2800, 5600, …
  (−∞, 0]nonnegativerational roots
• A053122 → D+1
  1 · -1, 1 · -1, -2, 1 · 6, -2, -3, 1 … Σ 1, 0, -2, 2, 1, -4, 3, 2, …
  (−∞, ∞)
• A038322: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
  1 · 11, 8 · 121, 176, 64 · 1331, 2904, 2112, 512 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  (−∞, −1]nonnegativerational roots
• A091044 → basis A007318 → x ↦ x/(x+1)
  1 · 4, 6 · 16, 48, 35 · 64, 288, 424, 204 … Σ 1, 10, 99, 980, 9701, 96030, 950599, …
  [−1, 0]nonnegative
• A081580 → D
  1 · 7, 2 · 13, 26, 3 · 19, 122, 57, 4 … Σ 0, 1, 9, 42, 202, 855, 3567, 14308, …
  (−∞, 0]nonnegative
• A143494 → (θ+1)²
  1 · 2, 4 · 4, 20, 9 · 8, 76, 81, 16 … Σ 1, 6, 33, 181, 1020, 5977, 36589, …
  (−∞, 0]nonnegative
• A091044 → basis A013610 → D+1
  1 · 10, 6 · 64, 102, 27 · 352, 1008, 684, 108 … Σ 1, 16, 193, 2152, 23065, 240808, …
  (−∞, 0]nonnegative
• A055151 → D → D
  4 · 20 · 60, 30 · 140, 210 … Σ 0, 0, 0, 0, 4, 20, 90, 350, …
  (−∞, 0]nonnegative
• A137597 → reverse → x ↦ x+1
  1 · 3, 2 · 11, 15, 5 · 47, 98, 67, 15 … Σ 1, 5, 31, 227, 1897, 17693, 181351, …
  (−∞, −1]nonnegative
• A038271 → x ↦ x/(x+1) → x ↦ x/(x+1)
  1 · 7, 19 · 49, 266, 361 · 343, 2793, 7581, 6859 … Σ 1, 26, 676, 17576, 456976, 11881376, …
  [−1, 0]nonnegativerational roots
• A156289 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 17, 31 · 1, 66, 339, 379 … Σ 1, 5, 49, 785, 18121, 563045, 22458409, …
  [−1, 0]nonnegative
• A156289 → θ+1
  1 · 1, 6 · 1, 30, 45 · 1, 126, 630, 420 … Σ 1, 7, 76, 1177, 24451, 650992, …
  (−∞, 0]nonnegative
• A136124 → basis A013609 → θ+1
  1 · 3, 4 · 12, 28, 12 · 60, 188, 144, 32 … Σ 1, 7, 52, 424, 3812, 37668, 406920, …
  (−∞, 0]nonnegative
• A099040 → basis A007318 → D+1
  1 · 4, 2 · 16, 18, 4 · 56, 104, 56, 8 … Σ 1, 6, 38, 224, 1280, 7136, 39072, …
  (−∞, 0]nonnegative
• A063967 → x ↦ x−4
  1 · -3, 1 · 6, -5, 1 · -9, 15, -7, 1 … Σ 1, -2, 2, 0, -4, 8, -8, 0, …
  (−1, 3]
• type_b_eulerian → γ (scale 1/4) → x ↦ x+1 → D+1
  1 · 1 · 3, 1 · 11, 5 … Σ 1, 1, 4, 16, 95, 663, 5590, 54218, …
  (−∞, 0]nonnegative
• A271703 → θ+1
  1 · 0, 2 · 0, 4, 3 · 0, 12, 18, 4 … Σ 1, 2, 7, 34, 209, 1546, 13327, 130922, …
  (−∞, 0]nonnegative
• A125092 → θ+1 → (θ+1)²
  1 · 1, 32 · 1, 64, 243 · 1, 96, 729, 1024 … Σ 1, 33, 308, 1850, 8808, 36232, 134656, …
  (−∞, 0]nonnegative
• A269944 → basis A013609 → D+1
  1 · 3, 2 · 8, 14, 4 · 44, 98, 56, 8 … Σ 1, 5, 26, 206, 2640, 52176, 1499232, …
  (−∞, 0]nonnegative
• A013610 → basis A013611 → x ↦ x+1
  1 · 16, 12 · 256, 384, 144 · 4096, 9216, 6912, 1728 … Σ 1, 28, 784, 21952, 614656, 17210368, …
  (−∞, −1]nonnegativerational roots
• A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → reverse
  1 · 1, 3 · 1, 7, 11 · 1, 13, 46, 49 … Σ 1, 4, 19, 109, 742, 5815, 51193, …
  (−∞, 0]nonnegative
• A069466 → θ+1
  1 · 2, 4 · 6, 48, 18 · 20, 360, 540, 80 … Σ 1, 6, 72, 1000, 14700, 222264, 3415104, …
  (−∞, 0]nonnegative
• A136157 → (θ+1)²
  3 · 1, 12 · 0, 4, 27 · 0, 0, 9, 48 … Σ 3, 13, 31, 57, 91, 133, 183, 241, …
  (−∞, 0]nonnegativerational roots
• A052553 → basis A013611 → D
  4 · 8 · 20, 32 · 40, 96 … Σ 0, 0, 4, 8, 52, 136, 560, 1624, …
  (−∞, 0]nonnegative
• A204420 → basis A007318 → D+1
  1 · 2, 1 · 21, 18, 3 · 570, 615, 180, 15 … Σ 1, 3, 42, 1380, 82320, 7771680, …
  (−∞, 0]nonnegative
• A048854 → D
  1 · 12, 2 · 180, 60, 3 · 3360, 1680, 168, 4 … Σ 0, 1, 14, 243, 5212, 133925, 4016106, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → x ↦ x+1 → (θ+1)²
  1 · -2, -4 · 4, 16, 9 · -8, -48, -54, -16 … Σ 1, -6, 29, -126, 513, -1998, 7533, …
  (−∞, 0]
• A166962 → x ↦ x+1
  1 · 2, 1 · 14, 14, 1 · 174, 244, 72, 1 … Σ 1, 3, 29, 491, 12893, 472947, 22873821, …
  (−∞, −1]nonnegative
• A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
  1 · 1, 2 · 1, 5, 6 · 1, 9, 26, 24 … Σ 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0)[−1, 0]nonnegative+1
• A064861 → D
  3, 4 · 5, 16, 12 · 6, 26, 36, 16 … Σ 7, 33, 84, 324, 756, 2700, 6048, 20736, …
  (−∞, 0]nonnegativerational roots
• A038326 → θ+1
  1 · 11, 24 · 121, 528, 432 · 1331, 8712, 14256, 6912 … Σ 1, 35, 1081, 31211, 863857, 23226803, …
  (−∞, 0]nonnegativerational roots
• A121574 → D+1
  1 · 3, 1 · 9, 7, 1 · 24, 32, 11, 1 … Σ 1, 4, 17, 68, 264, 1002, 3739, 13768, …
  (−∞, 0]nonnegative
• A001263 → basis A013611 → D
  4 · 20, 32 · 84, 288, 192 · 336, 1792, 2688, 1024 … Σ 0, 4, 52, 564, 5840, 59560, 604284, …
  (−∞, 0]nonnegative
• A038326 → basis A013609 → x ↦ x/(x+1)
  1 · 23, 47 · 529, 2162, 2209 … Σ 1, 70, 4900, 343000, 24010000, …
  [−1, 0]nonnegativerational roots
• A356546 → basis A013611 → D
  8 · 96, 192 · 960, 3840, 3840 … Σ 0, 8, 288, 8640, 241920, 6531840, …
  (−∞, 0]nonnegativerational roots
• A136521 → D
  2 · 0, 4 · 0, 0, 6 · 0, 0, 0, 8 … Σ 0, 2, 4, 6, 8, 10, 12, 14, …
  (−∞, 0]nonnegativerational roots
• A267006 → (θ+1)²
  1 · 0, 4, 9 · 0, 0, 0, 16, 25 · 0, 0, 0, 0, 0, 36, 49 … Σ 1, 13, 41, 85, 145, 221, 313, 421, …
  (−∞, 0]nonnegativerational roots
• A038326 → x ↦ x/(x+1)
  1 · 11, 23 · 121, 506, 529 · 1331, 8349, 17457, 12167 … Σ 1, 34, 1156, 39304, 1336336, 45435424, …
  [−1, 0]nonnegativerational roots
• A088996 → basis A013609 → x ↦ x/(x+1)
  1 · 1, 3 · 3, 16, 21 · 15, 113, 281, 231 … Σ 1, 4, 40, 640, 14080, 394240, 13404160, …
  [−1, 0]nonnegativerational roots
• A062145 → θ+1
  1 · 1, 8 · 1, 20, 30 · 1, 36, 135, 80 … Σ 1, 9, 51, 252, 1170, 5247, 23023, …
  (−∞, 0]nonnegative
• A274496 → basis A013611 → D
  8 · 16 · 64, 128 · 128, 256 … Σ 0, 0, 8, 16, 192, 384, 3456, 6912, …
  (−∞, 0]nonnegativerational roots
• A132774 → reverse → x ↦ x+1
  1 · 5, 2 · 9, 4 · 13, 6 … Σ 1, 7, 13, 19, 25, 31, 37, 43, …
  (−∞, −1]nonnegativerational roots
• A026729 → basis A013611 → reverse
  1 · 4, 1 · 16, 12, 2 · 64, 80, 28, 3 … Σ 1, 5, 30, 175, 1025, 6000, 35125, …
  (−∞, 0]nonnegative
• A100861 → basis A013610 → θ+1
  1 · 1 · 2, 6 · 4, 18 … Σ 1, 1, 8, 22, 163, 671, 5026, 26188, …
  (−∞, 0]nonnegative
• A063967 → reverse
  1 · 1, 1 · 1, 3, 2 · 1, 5, 7, 3 … Σ 1, 2, 6, 16, 44, 120, 328, 896, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → x ↦ x+1
  1 · 2 · 6, 2 · 24, 16 … Σ 1, 2, 8, 40, 256, 1952, 17408, 177280, …
  (−∞, −1]nonnegative
• A114656 → basis A013609 → x ↦ x+1
  1 · 5, 2 · 31, 24, 4 · 215, 246, 84, 8 … Σ 1, 7, 59, 553, 5541, 58107, 629799, …
  (−∞, −1]nonnegative