Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A128813 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 40, 44, 12 · 760, 1316, 756, 144 … Σ 1, 6, 96, 2976, 151776, 11534976, …
  (−∞, −1]nonnegativerational roots
• A035342 → (θ+1)²
  1 · 3, 4 · 15, 36, 9 · 105, 348, 162, 16 … Σ 1, 7, 60, 631, 7915, 115536, 1924867, …
  (−∞, 0]nonnegative
• A099097 → reverse → (θ+1)²
  1 · 3 · 9, 4 · 27, 24 … Σ 1, 3, 13, 51, 198, 756, 2851, 10641, …
  (−∞, 0]nonnegative
• delannoy → basis A013611 → θ+1
  1 · 2, 8 · 5, 40, 48 · 12, 144, 384, 256 … Σ 1, 10, 93, 796, 6509, 51582, 399665, …
  (−∞, 0]nonnegative
• A127648 → (θ+1)² → D
  8 · 0, 54 · 0, 0, 192 · 0, 0, 0, 500 … Σ 0, 0, 8, 54, 192, 500, 1080, 2058, …
  (−∞, 0]nonnegativerational roots
• A390433 → θ+1
  1 · 0, 2 · 0, 8, 3 · 0, 72, 60, 4 … Σ 1, 2, 11, 136, 2989, 102142, 4986679, …
  (−∞, 0]nonnegative
• A136124 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 17, 12 · 24, 98, 133, 60 … Σ 1, 5, 35, 315, 3465, 45045, 675675, …
  [−1, 0]nonnegativerational roots
• A038328 → basis A013611 → D
  224, 128 · 4704, 5376, 1536 … Σ 352, 11616, 340736, 9370240, 247374336, …
  (−∞, 0]nonnegativerational roots
• A038320 → basis A013611 → x ↦ x+1
  1681, 1968, 576 · 68921, 121032, 70848, 13824 … Σ 4225, 274625, 17850625, 1160290625, …
  (−∞, −1]nonnegativerational roots
• A154228 → basis A007318
  1 · 2, 1 · 18, 18, 1 · 96, 144, 50, 1 … Σ 1, 3, 37, 291, 4943, 67791, 1587413, …
  (−∞, −1]nonnegative
• A121574 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 13, 10 · 8, 40, 64, 33 … Σ 1, 5, 27, 145, 779, 4185, 22483, …
  [−1, 0]nonnegative
• basis-map coefficients of A112227 → x ↦ x+1 → basis A007318
  4, 4, 1 · 8, 12, 6, 1 · 16, 32, 24, 8, 1 … Σ 9, 27, 81, 243, 729, 2187, 6561, 19683, …
  (−∞, −1]nonnegativerational roots
• A269951 → θ+1
  1 · 0, 2 · 0, 4, 3 · 0, 10, 15, 4 … Σ 1, 2, 7, 29, 142, 814, 5391, 40689, …
  (−∞, 0]nonnegative
• A204420 → basis A013609 → D
  2 · 24, 24 · 690, 1080, 360 … Σ 0, 2, 48, 2130, 156240, 17253810, …
  (−∞, 0]nonnegative
• A272867 → x ↦ x+1
  1 · 6, 6, 1 · 36, 72, 48, 12, 1 … Σ 1, 13, 169, 2197, 28561, 371293, …
  (−∞, −1]nonnegative
• A063967 → x ↦ x−8
  1 · -7, 1 · 42, -13, 1 · -245, 119, -19, 1 … Σ 1, -6, 30, -144, 684, -3240, 15336, …
  (3, 7]
• A133080 → θ+1 → reverse
  1 · 2, 1 · 3 · 4, 3 … Σ 1, 3, 3, 7, 5, 11, 7, 15, …
  (−∞, 0]nonnegativerational roots
• A069466 → (θ+1)²
  1 · 2, 8 · 6, 96, 54 · 20, 720, 1620, 320 … Σ 1, 10, 156, 2680, 46900, 822024, …
  (−∞, 0]nonnegative
• A038334 → basis A013609 → x ↦ x/(x+1)
  400, 1440, 1296 · 8000, 43200, 77760, 46656 … Σ 3136, 175616, 9834496, 550731776, …
  [−1, 0]nonnegativerational roots
• A225470 → basis A013610 → (θ+1)²
  1 · 3, 12 · 18, 108, 81 · 162, 1188, 1458, 432 … Σ 1, 15, 207, 3240, 58644, 1217916, …
  (−∞, 0]nonnegative
• A290315 → θ+1
  1 · 1, 4 · 1, 32, 36 · 1, 132, 852, 480 … Σ 1, 5, 69, 1465, 41273, 1441757, …
  (−∞, 0]nonnegative
• A290319 → basis A013609 → reverse
  1 · 2, 2 · 4, 16, 12 · 8, 72, 184, 120 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
  (−∞, 0]nonnegativerational roots
• A207607: Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  1 · 1, 2 · 1, 5, 2 · 1, 9, 9, 2 … Σ 1, 3, 8, 21, 55, 144, 377, 987, …
  (−4, 0)nonnegative
• A034870 → Veronese 2 → (θ+1)²
  1 · 1, 4 · 1, 24, 9 · 1, 60, 135, 16 … Σ 1, 5, 34, 212, 1216, 6592, 34304, …
  (−∞, 0]nonnegative
• A038305 → D
  60, 18 · 900, 540, 81 · 12000, 10800, 3240, 324 … Σ 78, 1521, 26364, 428415, 6683274, …
  (−∞, 0]nonnegativerational roots
• A038231 → Veronese 2 → D
  1 · 12 · 96, 2 · 640, 40 … Σ 0, 0, 1, 12, 98, 680, 4323, 26068, …
  (−∞, 0]nonnegative
• A094638 → D → D
  4 · 22, 36 · 70, 300, 288 · 170, 1350, 3288, 2400 … Σ 0, 0, 4, 58, 658, 7208, 81128, 959992, …
  (−∞, 0]nonnegative
• A102365 → x ↦ x+1 → D+1
  1 · 1 · 3, 1 · 14, 9, 1 … Σ 1, 1, 4, 24, 195, 1995, 24570, 353430, …
  (−∞, 0]nonnegative
• A133080 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 1, 6, 9 · 2, 16, 42, 36 … Σ 1, 6, 16, 96, 256, 1536, 4096, 24576, …
  [−1, 0]nonnegativerational roots
• A038284 → basis A013609 → x ↦ x/(x+1)
  196, 728, 676 · 2744, 15288, 28392, 17576 … Σ 1600, 64000, 2560000, 102400000, …
  [−1, 0]nonnegativerational roots
• A008288 → basis A013609 → reverse
  1 · 2, 2 · 4, 10, 5 · 8, 32, 36, 12 … Σ 1, 4, 19, 88, 409, 1900, 8827, 41008, …
  (−∞, 0]nonnegative
• A257614 → D+1
  1 · 4, 2 · 32, 36, 4 · 252, 732, 268, 8 … Σ 1, 6, 72, 1260, 28728, 804384, …
  (−∞, 0]nonnegative
• A110552 → θ+1
  1 · 1, 4 · 1, 10, 12 · 1, 20, 51, 32 … Σ 1, 5, 23, 104, 468, 2096, 9328, 41216, …
  (−∞, 0]nonnegative
• A144699 → x ↦ x+1
  1 · 6, 5 · 42, 66, 25 · 336, 756, 546, 125 … Σ 1, 11, 133, 1763, 25525, 401891, …
  (−∞, −1]nonnegative
• A121314 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 4, 27, 16 … Σ 1, 4, 13, 47, 170, 593, 1997, 6536, …
  (−∞, 0]nonnegative
• A030528 → x ↦ x/(x+1)
  1 · 1, 2 · 0, 2, 3 · 0, 1, 5, 5 … Σ 1, 3, 5, 11, 21, 43, 85, 171, …
  [−1, 0]nonnegative
• A249251 → basis A007318
  1 · 4, 3 · 13, 18, 6 · 43, 87, 57, 12 … Σ 1, 7, 37, 199, 1069, 5743, 30853, …
  (−∞, −1]nonnegative
• A038271 → basis A013611 → θ+1
  1 · 12, 40 · 144, 960, 1200 … Σ 1, 52, 2304, 94208, 3670016, 138412032, …
  (−∞, 0]nonnegativerational roots
• A123519 → (θ+1)²
  1, 24, 36 · 1, 48, 180, 128 · 1, 80, 540, 896, 400 … Σ 61, 357, 1917, 9717, 47249, 222633, …
  (−∞, 0]nonnegative
• A257617 → D+1
  1 · 4, 2 · 40, 44, 4 · 396, 1164, 412, 8 … Σ 1, 6, 88, 1980, 59400, 2217600, …
  (−∞, 0]nonnegative
• A269939 → (θ+1)²
  1 · 0, 4 · 0, 4, 27 · 0, 4, 90, 240 … Σ 1, 4, 31, 334, 4534, 73868, 1401068, …
  (−∞, 0]nonnegative
• A152198 → basis A013610 → D
  3 · 3 · 12, 18 · 12, 18 … Σ 0, 0, 3, 3, 30, 30, 225, 225, …
  (−∞, 0]nonnegativerational roots
• A113214 → θ+1 → reverse
  1 · 2, 2 · 3, 6 · 4, 12, 4 … Σ 1, 4, 9, 20, 40, 78, 147, 272, …
  (−∞, 0]nonnegative
• A267482 → x ↦ x+1
  1 · 2, 1 · 1, 3, 1 · -1, 3, 4, 1 … Σ 1, 3, 5, 7, 9, 11, 13, 15, …
  (−3, 1)
• A008288 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 5, 20, 19 · 12, 72, 140, 88 … Σ 1, 6, 44, 312, 2224, 15840, 112832, …
  [−1, 0]nonnegative
• A085881 → basis A013610
  1 · 2, 3 · 12, 36, 27 · 120, 540, 810, 405 … Σ 1, 5, 75, 1875, 65625, 2953125, …
  [−1, 0]nonnegativerational roots
• delannoy → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 15, 4 … Σ 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0]nonnegative
• A038331 → basis A013611 → D+1
  969, 1480, 400 · 22253, 58140, 44400, 8000 … Σ 2849, 132793, 5926401, 256760057, …
  (−∞, 0]nonnegativerational roots
• A028246 → D
  1 · 3, 4 · 7, 24, 18 · 15, 100, 180, 96 … Σ 0, 1, 7, 49, 391, 3601, 37927, 451249, …
  (−∞, 0]nonnegative
• A052553 → basis A013609 → D+1
  1 · 1 · 4, 2 · 7, 4 … Σ 1, 1, 6, 11, 37, 84, 233, 565, …
  (−∞, 0]nonnegative