Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A178822 → x ↦ x+1 → (θ+1)²
  1 · 12, 24 · 84, 336, 189 · 448, 2688, 3024, 896 … Σ 1, 36, 609, 7056, 64638, 503496, …
  (−∞, 0]nonnegative
• A143495 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 4, 5 · 17, 43, 27 · 77, 295, 374, 157 … Σ 1, 9, 87, 903, 10041, 119217, 1505679, …
  [−1, 0]nonnegative
• A099040 → x ↦ x−3
  1 · -6, 2 · 30, -22, 4 · -144, 168, -64, 8 … Σ 1, -4, 12, -32, 80, -192, 448, -1024, …
  (1, 3]
• A173018 → reverse (shift +1) → reverse → D
  1 · 4, 2 · 11, 22, 3 · 26, 132, 78, 4 … Σ 0, 0, 1, 6, 36, 240, 1800, 15120, …
  (−∞, 0]nonnegative
• A038336 → basis A013609 → θ+1
  484, 1760, 1200 · 10648, 58080, 79200, 32000 … Σ 3444, 179928, 9038736, 441860832, …
  (−∞, 0]nonnegativerational roots
• A371081 → (θ+1)²
  1 · 16, 4 · 400, 208, 9 · 14400, 11792, 1044, 16 … Σ 1, 20, 617, 27252, 1632465, 127338092, …
  (−∞, 0]nonnegative
• A027466 → reverse
  1 · 1, 7 · 1, 14, 49 · 1, 21, 147, 343 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, 0]nonnegativerational roots
• A103633 → basis A007318 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 2, 11, 20, 12 … Σ 1, 3, 15, 45, 225, 675, 3375, 10125, …
  [−1, 0]nonnegativerational roots
• A178822 → Veronese 2 → θ+1
  1 · 6 · 21, 42 · 56, 336 … Σ 1, 6, 63, 392, 2016, 9072, 36960, …
  (−∞, 0]nonnegative
• A038281 → basis A013611 → (θ+1)²
  121, 1056, 1296 · 1331, 17424, 42768, 27648 … Σ 2473, 89171, 2946001, 91848683, …
  (−∞, 0]nonnegative
• A159854 → x ↦ x+1 → x ↦ x+1
  4, 4, 1 · 12, 16, 7, 1 · 36, 60, 37, 10, 1 … Σ 9, 36, 144, 576, 2304, 9216, 36864, …
  (−∞, −1]nonnegativerational roots
• A088996 → basis A013610 → reverse
  1 · 3, 1 · 18, 15, 3 · 162, 225, 102, 15 … Σ 1, 4, 36, 504, 9576, 229824, 6664896, …
  (−∞, 0]nonnegativerational roots
• A108426 → D+1
  1 · 2, 1 · 8, 9, 2 · 40, 70, 36, 5 … Σ 1, 3, 19, 151, 1331, 12427, 120275, …
  (−∞, 0]nonnegative
• basis-map coefficients of A371080, shifted → basis A013610 → reverse
  1 · 3, 1 · 9, 15, 4 · 27, 108, 117, 28 … Σ 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0]nonnegativerational roots
• A145324 → basis A007318 → θ+1
  1 · 3, 4 · 12, 34, 18 · 60, 266, 294, 96 … Σ 1, 7, 64, 716, 9451, 143653, 2470110, …
  (−∞, 0]nonnegative
• A272866 → D
  3, 2 · 6, 22, 18, 4 · 9, 60, 135, 120, 45, 6 … Σ 0, 5, 50, 375, 2500, 15625, 93750, …
  (−∞, 0]nonnegative
• A038305 → (θ+1)²
  100, 240, 81 · 1000, 3600, 2430, 432 … Σ 421, 7462, 125905, 2051998, 32588101, …
  (−∞, 0]nonnegative
• A038208 → Veronese 2 (j=1) → (θ+1)²
  2 · 8 · 24, 32 · 64, 256 … Σ 0, 2, 8, 56, 320, 1728, 8960, 45056, …
  (−∞, 0]nonnegative
• A008459 → x ↦ x+1 → D
  1 · 6, 2 · 30, 24, 3 · 140, 180, 60, 4 … Σ 0, 1, 8, 57, 384, 2505, 16008, 100849, …
  (−∞, −1]nonnegative
• A143683 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 40, 60, 22, 1 … Σ 1, 3, 25, 123, 769, 4275, 25129, …
  (−∞, −1]nonnegative
• narayana → basis A007318 → D+1
  1 · 3, 1 · 10, 7, 1 · 35, 39, 12, 1 … Σ 1, 4, 18, 87, 439, 2278, 12052, 64669, …
  (−∞, 0]nonnegative
• A104978 → x ↦ x+1
  1 · 2, 1 · 10, 11, 3 · 66, 113, 64, 12 … Σ 1, 3, 24, 255, 3102, 40854, 566934, …
  (−∞, −1]nonnegative
• A172431 → basis A007318 → D
  2 · 10, 6 · 38, 44, 12 · 130, 222, 120, 20 … Σ 0, 2, 16, 94, 492, 2426, 11528, 53398, …
  (−∞, −1]nonnegative
• A023531 → Veronese 2 → D+1
  1 · 1, 1 · 0, 2, 1 · 0, 0, 3, 1 … Σ 1, 0, 2, 0, 3, 0, 4, 0, …
  (−∞, 0]nonnegativerational roots
• A372868: veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
  1 · 2 · 4, 1 · 8, 6 … Σ 1, 2, 5, 14, 41, 122, 365, 1094, …
  (−∞, 0)nonnegative
• A131689 → (θ+1)² → reverse
  1 · 4 · 18, 4 · 96, 54, 4 … Σ 1, 4, 22, 154, 1306, 12994, 148282, …
  (−∞, 0]nonnegative
• A110441 → basis A013610 → θ+1
  1 · 4, 6 · 14, 48, 27 · 48, 264, 324, 108 … Σ 1, 10, 89, 744, 5975, 46670, 357181, …
  (−∞, 0]nonnegative
• A249252 → reverse → x ↦ x/(x+1)
  1 · 4, 5 · 8, 24, 17 · 16, 80, 124, 61 … Σ 1, 9, 49, 281, 1601, 9129, 52049, …
  [−1, 0]nonnegative
• A274496 → basis A013611 → D+1
  1 · 2 · 12, 8 · 24, 16 … Σ 1, 2, 20, 40, 336, 672, 5184, 10368, …
  (−∞, 0]nonnegativerational roots
• A122848 → D
  1 · 1, 2 · 0, 6, 3 · 0, 6, 18, 4 … Σ 0, 1, 3, 9, 28, 90, 306, 1078, …
  (−∞, 0]nonnegative
• A225470 → x ↦ x+1
  1 · 3, 1 · 18, 9, 1 · 162, 99, 18, 1 … Σ 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, −1]nonnegativerational roots
• A114687 → basis A007318
  1 · 3, 2 · 11, 14, 4 · 45, 84, 48, 8 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, −1]nonnegative
• A086645 → basis A013610 → D
  3 · 24, 18 · 144, 324, 81 · 768, 2880, 2592, 324 … Σ 0, 3, 42, 549, 6564, 73815, 797166, …
  (−∞, 0]nonnegative
• A124932 → basis A007318 → reverse
  1 · 3, 5 · 6, 21, 18 · 10, 54, 96, 56 … Σ 1, 8, 45, 216, 945, 3888, 15309, 58320, …
  (−∞, 0]nonnegativerational roots
• A038226 → basis A007318 → reverse
  64, 176, 121 · 512, 2112, 2904, 1331 … Σ 361, 6859, 130321, 2476099, 47045881, …
  (−∞, 0]nonnegativerational roots
• A038277 → basis A013609 → θ+1
  1 · 18, 44 · 324, 1584, 1452 … Σ 1, 62, 3360, 169600, 8192000, …
  (−∞, 0]nonnegativerational roots
• A166973 → (θ+1)²
  1 · 1, 4 · 1, 28, 9 · 1, 172, 162, 16 … Σ 1, 5, 38, 351, 3775, 46382, 638395, …
  (−∞, 0]nonnegative
• delannoy → x ↦ x+1 → θ+1
  1 · 2, 2 · 5, 10, 3 · 12, 36, 24, 4 … Σ 1, 4, 18, 76, 314, 1272, 5078, 20036, …
  (−∞, 0]nonnegative
• A152198 → θ+1
  1 · 1 · 1, 2 · 1, 2 … Σ 1, 1, 3, 3, 8, 8, 20, 20, …
  (−∞, 0]nonnegativerational roots
• A257626 → x ↦ x+1
  1 · 12, 6 · 180, 180, 36 · 3240, 4860, 2052, 216 … Σ 1, 18, 396, 10368, 315576, 10958328, …
  (−∞, −1]nonnegative
• A064861 → x ↦ x+1 → x ↦ x/(x+1)
  6, 19, 15 · 18, 87, 140, 75 · 36, 228, 541, 570, 225 … Σ 40, 320, 1600, 12800, 64000, 512000, …
  [−1, 0]nonnegativerational roots
• A026374 → D+1 → x ↦ x+1
  1 · 3, 1 · 10, 7, 1 · 25, 29, 10, 1 … Σ 1, 4, 18, 65, 275, 946, 3872, 12947, …
  (−∞, −1]nonnegative
• A113953 → basis A013610 → x ↦ x+1
  1 · 4, 3 · 24, 30, 9 · 128, 240, 144, 27 … Σ 1, 7, 63, 539, 4655, 40131, 346087, …
  (−∞, −1]nonnegative
• A124932 → D
  3 · 9, 12 · 18, 48, 30 · 30, 120, 150, 60 … Σ 0, 3, 21, 96, 360, 1200, 3696, 10752, …
  (−∞, 0]nonnegativerational roots
• A094638 → D+1 → D
  1 · 7, 4 · 28, 58, 18 · 80, 370, 438, 96 … Σ 0, 1, 11, 104, 984, 9764, 103340, …
  (−∞, 0]nonnegative
• A038284 → basis A013610 → θ+1
  196, 1008, 972 · 2744, 21168, 40824, 23328 … Σ 2176, 88064, 3407872, 127926272, …
  (−∞, 0]nonnegativerational roots
• A161198 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 7 · 15, 62, 63 · 105, 599, 1123, 693 … Σ 1, 10, 140, 2520, 55440, 1441440, …
  [−1, 0]nonnegativerational roots
• A127648 → D → (θ+1)²
  2 · 0, 24 · 0, 0, 108 · 0, 0, 0, 320 … Σ 0, 0, 2, 24, 108, 320, 750, 1512, …
  (−∞, 0]nonnegativerational roots
• A120910 → basis A013611 → reverse
  3 · 12, 9 · 48, 72, 27 · 192, 432, 324, 81 … Σ 3, 21, 147, 1029, 7203, 50421, 352947, …
  (−∞, 0]nonnegativerational roots
• A391533 → basis A013609 → θ+1
  9, 24, 12 · 18, 84, 96, 32 · 36, 240, 444, 320, 80 … Σ 45, 230, 1120, 5280, 24320, 110080, …
  (−∞, 0]nonnegative