Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A013613 → x ↦ x+1 → D+1
  1 · 13, 6 · 133, 156, 36 · 1225, 2394, 1404, 216 … Σ 1, 19, 325, 5239, 81289, 1228123, …
  (−∞, 0]nonnegativerational roots
• A118931 → basis A013609 → x ↦ x+1
  1 · 1 · 1 · 7, 4 … Σ 1, 1, 1, 11, 41, 101, 1201, 7351, …
  (−∞, −1]nonnegative
• A115951 → D
  2 · 2, 12 · 0, 24, 60 · 0, 12, 180, 280 … Σ 0, 2, 14, 84, 472, 2560, 13584, 71008, …
  (−∞, 0]nonnegative
• A038303 → basis A013610 → θ+1
  1 · 11, 6 · 121, 132, 27 · 1331, 2178, 891, 108 … Σ 1, 17, 280, 4508, 71344, 1114064, …
  (−∞, 0]nonnegativerational roots
• A028326 → D
  2 · 4, 4 · 6, 12, 6 · 8, 24, 24, 8 … Σ 0, 2, 8, 24, 64, 160, 384, 896, …
  (−∞, 0]nonnegativerational roots
• A038321 → basis A013611 → basis A007318
  2116, 2576, 784 · 97336, 177744, 108192, 21952 … Σ 5476, 405224, 29986576, 2219006624, …
  (−∞, −1]nonnegativerational roots
• image under θ+1 → θ+1 → D+1
  1 · 4, 4 · 0, 18, 9 · 0, 0, 48, 16 … Σ 1, 8, 27, 64, 125, 216, 343, 512, …
  (−∞, 0]nonnegativerational roots
• A144696 → D+1
  1 · 3, 2 · 8, 15, 4 · 19, 84, 57, 8 … Σ 1, 5, 27, 168, 1200, 9720, 88200, …
  (−∞, 0]nonnegative
• A145324 → basis A013611 → D+1
  1 · 11, 8 · 80, 260, 96 · 592, 3668, 6176, 1536 … Σ 1, 19, 436, 11972, 385192, 14247256, …
  (−∞, 0]nonnegative
• A062196 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 10, 15 · 1, 18, 63, 56 … Σ 1, 5, 26, 138, 743, 4043, 22180, …
  [−1, 0]nonnegative
• A133087 → (θ+1)²
  1 · 2, 4 · 1, 8, 9 · 2, 20, 36, 16 … Σ 1, 6, 18, 74, 160, 592, 1120, 3936, …
  (−∞, 0]nonnegative
• A132440 → D+1 → (1/n)D²+(2/n)D+1/n → x ↦ x+1
  1 · 4, 1 · 13, 8, 1 · 34, 39, 12, 1 … Σ 1, 5, 22, 86, 304, 992, 3040, 8864, …
  (−∞, −1]nonnegative
• A124324 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 4 … Σ 1, 1, 3, 9, 35, 153, 755, 4105, …
  (−∞, −1]nonnegative
• A127648 → Veronese 2 → θ+1
  1 · 0, 6 · 0, 0, 15 · 0, 0, 0, 28 … Σ 0, 1, 0, 6, 0, 15, 0, 28, …
  (−∞, 0]nonnegativerational roots
• A038284 → basis A013609 → reverse
  144, 336, 196 · 1728, 6048, 7056, 2744 … Σ 676, 17576, 456976, 11881376, …
  (−∞, 0]nonnegativerational roots
• A199478 → D+1
  2, 1 · 4, 3 · 6, 9, 2 · 8, 23, 8 … Σ 3, 7, 17, 39, 89, 199, 441, 967, …
  (−∞, 0]nonnegative
• A261711 → D
  1 · 8 · 46, 2 · 232, 24 … Σ 0, 0, 1, 8, 48, 256, 1280, 6144, …
  (−∞, 0]nonnegative
• A011973 → θ+1
  1 · 1 · 1, 2 · 1, 4 … Σ 1, 1, 3, 5, 10, 18, 33, 59, …
  (−∞, 0]nonnegative
• A038291 → basis A013609 → D+1
  1 · 12, 2 · 140, 48, 4 · 1600, 840, 144, 8 … Σ 1, 14, 192, 2592, 34560, 456192, …
  (−∞, 0]nonnegativerational roots
• A105868 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 2, 3 · 0, 0, 6, 7 … Σ 1, 1, 5, 13, 49, 161, 581, 2045, …
  [−1, 0]nonnegative
• A163936 → reverse → D+1
  1 · 1, 1 · 1, 5, 2 · 1, 17, 26, 6 … Σ 1, 2, 8, 50, 420, 4410, 55440, 810810, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 (shift +1) → reverse → θ+1
  1 · 1, 4 · 1, 10, 15 · 1, 20, 69, 60 … Σ 1, 5, 26, 150, 963, 6807, 52448, …
  (−∞, 0]nonnegative
• A059364 → basis A007318 → x ↦ x/(x+1)
  1 · 3, 4 · 15, 41, 28 · 105, 437, 606, 280 … Σ 1, 7, 84, 1428, 31416, 848232, …
  [−1, 0]nonnegativerational roots
• A085478 → D → D+1
  1 · 5, 2 · 16, 16, 3 · 40, 72, 33, 4 … Σ 0, 0, 1, 7, 35, 149, 576, 2088, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → reverse → (θ+1)²
  1 · 2, 4 · 3, 24, 9 · 4, 72, 126, 16 … Σ 1, 6, 36, 218, 1345, 8552, 56371, …
  (−∞, 0]nonnegative
• A038303 → basis A013610 → x ↦ x/(x+1)
  1 · 11, 14 · 121, 308, 196 · 1331, 5082, 6468, 2744 … Σ 1, 25, 625, 15625, 390625, 9765625, …
  [−1, 0]nonnegativerational roots
• A063967 → reverse → θ+1
  1 · 1, 2 · 1, 6, 6 · 1, 10, 21, 12 … Σ 1, 3, 13, 44, 148, 476, 1500, 4640, …
  (−∞, 0]nonnegative
• delannoy → basis A013610 → D
  3 · 15, 18 · 54, 144, 81 · 174, 720, 891, 324 … Σ 0, 3, 33, 279, 2109, 14979, 102249, …
  (−∞, 0]nonnegative
• A038330 → basis A013610 → basis A007318
  784, 672, 144 · 21952, 28224, 12096, 1728 … Σ 1600, 64000, 2560000, 102400000, …
  (−∞, −1]nonnegativerational roots
• A034807 → D+1
  2 · 1 · 3, 2 · 4, 3 … Σ 2, 1, 5, 7, 15, 26, 48, 85, …
  (−∞, 0]nonnegative
• A021009: Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).
  A123202 → x ↦ x/(x+1)
  1 · 1, -1 · 2, -4, 1 · 6, -18, 9, -1 … Σ 1, 0, -1, -4, -15, -56, -185, -204, …
  (−∞, ∞)
• A013610 → basis A013611 → reverse
  1 · 12, 4 · 144, 96, 16 · 1728, 1728, 576, 64 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, −1]nonnegativerational roots
• A136124 → (θ+1)²
  1 · 2, 4 · 6, 20, 9 · 24, 104, 81, 16 … Σ 1, 6, 35, 225, 1624, 13132, 118124, …
  (−∞, 0]nonnegative
• A143495 → D+1 → reverse
  1 · 1, 4 · 1, 9, 16 · 1, 15, 61, 64 … Σ 1, 5, 26, 141, 799, 4736, 29371, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x/(x+1)
  1 · 0, 3 · 0, 3, 12 · 0, 0, 18, 45 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  [−1, 0]nonnegative
• A104684 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 13, 50, 46 · 63, 360, 666, 396 … Σ 1, 9, 109, 1485, 21345, 316089, …
  [−1, 0]nonnegative
• A088996 → basis A013611 → x ↦ x/(x+1)
  1 · 1, 5 · 3, 26, 55 · 15, 181, 717, 935 … Σ 1, 6, 84, 1848, 55440, 2106720, …
  [−1, 0]nonnegativerational roots
• A116414 → basis A013611 → θ+1
  1 · 5, 8 · 22, 80, 48 · 95, 552, 720, 256 … Σ 1, 13, 150, 1623, 16857, 170208, …
  (−∞, 0]nonnegative
• A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.
  1 · 2 · 3, 1 · 4, 4 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 0)nonnegativeA202064
• A034870 → basis A013611 → x ↦ x+1
  1 · 36, 48, 16 · 1296, 3456, 3456, 1536, 256 … Σ 1, 100, 10000, 1000000, 100000000, …
  (−∞, −1]nonnegativerational roots
• A099040 → x ↦ x−7
  1 · -14, 2 · 182, -54, 4 · -2352, 1064, -160, 8 … Σ 1, -12, 132, -1440, 15696, -171072, …
  (5, 7]
• A038220 → D
  2 · 12, 8 · 54, 72, 24 · 216, 432, 288, 64 … Σ 0, 2, 20, 150, 1000, 6250, 37500, …
  (−∞, −1]nonnegativerational roots
• A274496 → x ↦ x+1 → x ↦ x+1
  1 · 2 · 6, 2 · 12, 4 … Σ 1, 2, 8, 16, 64, 128, 512, 1024, …
  (−∞, −1]nonnegativerational roots
• A059344 → θ+1
  1 · 1 · 1, 4 · 1, 12 … Σ 1, 1, 5, 13, 61, 221, 1081, 4705, …
  (−∞, 0]nonnegative
• A120910 → basis A007318 → (θ+1)²
  3 · 9, 12 · 27, 72, 27 · 81, 324, 243, 48 … Σ 3, 21, 126, 696, 3648, 18432, 90624, …
  (−∞, 0]nonnegative
• A136124 → basis A008277 → D
  1 · 8, 2 · 60, 30, 3 · 480, 360, 72, 4 … Σ 0, 1, 10, 93, 916, 9805, 114486, …
  (−∞, 0]nonnegative
• A391533 → Pₙ + Pₙ₊₁ → θ+1
  8, 24, 18, 4 · 8, 40, 54, 28, 5 … Σ 54, 135, 324, 756, 1728, 3888, 8640, …
  (−∞, 0]nonnegative
• A079638 → reverse
  1 · 1, 3 · 1, 9, 14 · 1, 18, 83, 90 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, 0]nonnegative
• A123519 → basis A013609 → D+1
  39, 60, 16 · 193, 504, 368, 64 … Σ 115, 1129, 10649, 97699, 878299, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → D
  2 · 6, 6 · 14, 36, 12 · 30, 150, 120, 20 … Σ 0, 0, 2, 12, 62, 320, 1712, 9604, …
  (−∞, 0]nonnegative