Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A207607 → basis A007318 → reverse
  1 · 2, 3 · 2, 9, 8 · 2, 15, 33, 21 … Σ 1, 5, 19, 71, 265, 989, 3691, 13775, …
  (−1, 0]nonnegative
• A122542 → basis A013610 → D+1
  1 · 4, 3 · 15, 30, 9 · 46, 165, 144, 27 … Σ 1, 7, 54, 382, 2606, 17294, 112494, …
  (−∞, 0]nonnegative
• A100861 → D → x ↦ x+1
  1 · 3 · 12, 6 · 40, 30 … Σ 0, 0, 1, 3, 18, 70, 375, 1701, …
  (−∞, −1]nonnegative
• A272867 → basis A013611 → D
  24, 32 · 288, 1536, 2304, 1024 … Σ 0, 56, 5152, 355488, 21803264, …
  (−∞, 0]nonnegative
• A008459 → basis A013610 → (θ+1)²
  1 · 2, 12 · 6, 72, 81 · 20, 360, 972, 432 … Σ 1, 14, 159, 1784, 19705, 213390, …
  (−∞, 0]nonnegative
• A013622 → basis A007318 → θ+1
  1 · 8, 10 · 64, 160, 75 · 512, 1920, 1800, 500 … Σ 1, 18, 299, 4732, 72501, 1085318, …
  (−∞, 0]nonnegativerational roots
• A038301 → θ+1
  81, 396, 363 · 729, 5346, 9801, 5324 … Σ 840, 21200, 512000, 12000000, …
  (−∞, 0]nonnegativerational roots
• A034870 → basis A013609 → D+1
  1 · 12, 16, 4 · 80, 256, 288, 128, 16 … Σ 1, 32, 768, 16384, 327680, 6291456, …
  (−∞, 0]nonnegativerational roots
• A152830 → x ↦ x+1 → basis A007318
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 4, 4, 4, 4, …
  (−∞, −1]nonnegativerational roots
• A257614 → D
  2 · 28, 8 · 244, 488, 24 · 1844, 11712, 5532, 64 … Σ 0, 2, 36, 756, 19152, 574560, 19994688, …
  (−∞, 0]nonnegative
• A038271 → reverse
  1 · 5, 7 · 25, 70, 49 · 125, 525, 735, 343 … Σ 1, 12, 144, 1728, 20736, 248832, …
  (−∞, 0]nonnegativerational roots
• A135065 → x ↦ x+1 → θ+1
  1 · 8, 8 · 36, 72, 27 · 128, 384, 288, 64 … Σ 1, 16, 135, 864, 4725, 23328, 107163, …
  (−∞, 0]nonnegativerational roots
• A008292 → γ (scale 1/2) (shift +1) → θ+1
  1 · 1 · 1, 2 · 1, 8 … Σ 1, 1, 3, 9, 35, 155, 791, 4529, …
  (−∞, 0]nonnegative
• A391533 → x ↦ x+1 → x ↦ x+1
  16, 8, 1 · 48, 40, 11, 1 · 144, 168, 73, 14, 1 … Σ 25, 100, 400, 1600, 6400, 25600, …
  (−∞, −1]nonnegativerational roots
• A156992 → basis A013611 → (θ+1)²
  1 · 4, 32 · 24, 384, 864 · 192, 4608, 20736, 24576 … Σ 1, 36, 1272, 50112, 2229120, 111352320, …
  (−∞, 0]nonnegative
• A091042: Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
  1 · 1, 3 · 1, 10, 5 · 1, 21, 35, 7 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0)nonnegative
• A161198 → basis A013611 → θ+1
  1 · 3, 16 · 15, 128, 192 · 105, 1136, 2880, 2048 … Σ 1, 19, 335, 6169, 122033, 2610347, …
  (−∞, 0]nonnegative
• A122542 → x ↦ x+1 → θ+1
  1 · 1, 2 · 3, 8, 3 · 7, 26, 21, 4 … Σ 1, 3, 14, 58, 238, 958, 3806, 14958, …
  (−∞, 0]nonnegative
• A306364 → x ↦ x+1
  1 · 3 · 15, 3 · 105, 45 … Σ 1, 3, 18, 150, 1620, 21420, 335160, …
  (−∞, −1]nonnegative
• A038220 → Veronese 2 → θ+1
  1 · 3 · 9, 8 · 27, 72 … Σ 1, 3, 17, 99, 561, 3123, 17185, 93747, …
  (−∞, 0]nonnegative
• A308737 → θ+1
  1 · 1, 6 · 1, 16, 21 · 1, 34, 93, 60 … Σ 1, 7, 38, 188, 888, 4080, 18400, 81856, …
  (−∞, 0]nonnegative
• A193685: 5-Stirling numbers of the second kind.
  1 · 5, 1 · 25, 11, 1 · 125, 91, 18, 1 … Σ 1, 6, 37, 235, 1540, 10427, 73013, …
  (−∞, 0)nonnegative
• A257618 → D+1
  1 · 4, 2 · 44, 48, 4 · 480, 1416, 496, 8 … Σ 1, 6, 96, 2400, 80640, 3386880, …
  (−∞, 0]nonnegative
• A274496 → x ↦ x+1 → D+1
  1 · 2 · 6, 2 · 12, 4 … Σ 1, 2, 8, 16, 60, 120, 432, 864, …
  (−∞, 0]nonnegativerational roots
• A054142 → reverse → D
  1 · 3, 2 · 6, 10, 3 · 10, 30, 21, 4 … Σ 0, 1, 5, 19, 65, 210, 654, 1985, …
  (−∞, 0]nonnegative
• A257615 → θ+1
  1 · 5, 10 · 25, 140, 75 · 125, 1430, 2145, 500 … Σ 1, 15, 240, 4200, 80640, 1693440, …
  (−∞, 0]nonnegative
• A120910 → basis A013610 → D
  9 · 54, 54 · 243, 486, 243 · 972, 2916, 2916, 972 … Σ 0, 9, 108, 972, 7776, 58320, 419904, …
  (−∞, 0]nonnegativerational roots
• A060920 → Veronese 2 → x ↦ x+1
  1 · 2 · 6, 1 · 22, 9 … Σ 1, 2, 7, 31, 140, 635, 2885, 13114, …
  (−∞, −1]nonnegative
• A257626 → x ↦ x/(x+1)
  1 · 6, 12 · 36, 180, 180 · 216, 2052, 4860, 3240 … Σ 1, 18, 396, 10368, 315576, 10958328, …
  [−1, 0]nonnegative
• A081580 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 9, 54, 61 · 28, 252, 676, 516 … Σ 1, 8, 124, 1472, 19216, 242048, …
  [−1, 0]nonnegative
• A159854 → basis A013611 → reverse
  16, 8, 1 · 64, 64, 20, 2 · 256, 384, 208, 48, 4 … Σ 25, 150, 900, 5400, 32400, 194400, …
  (−∞, 0]nonnegativerational roots
• A038329 → basis A013610 → x ↦ x/(x+1)
  225, 720, 576 · 3375, 16200, 25920, 13824 … Σ 1521, 59319, 2313441, 90224199, …
  [−1, 0]nonnegativerational roots
• A133087 → Veronese 2 (j=1) → D
  1 · 4 · 14, 2 · 20, 12 … Σ 0, 0, 0, 1, 4, 16, 32, 112, …
  (−∞, 0]nonnegative
• A133081 → reverse (shift +1) → x ↦ x/(x+1)
  1 · 0, 1, 2 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 1, 2 … Σ 1, 3, 1, 3, 1, 3, 1, 3, …
  [−1, 0]nonnegativerational roots
• A130534 → basis A013610 → (θ+1)²
  1 · 2, 12 · 6, 60, 81 · 24, 312, 729, 432 … Σ 1, 14, 147, 1497, 15792, 176436, …
  (−∞, 0]nonnegative
• A100861 → D → D+1
  1 · 3 · 12, 6 · 40, 30 … Σ 0, 0, 1, 3, 18, 70, 330, 1386, …
  (−∞, 0]nonnegative
• A106246 → D → (θ+1)²
  1 · 4, 8 · 3, 48, 27 · 0, 48, 216, 64 … Σ 0, 1, 12, 78, 328, 1035, 2676, 5992, …
  (−∞, 0]nonnegative
• A116414 → basis A013611 → D+1
  1 · 9, 4 · 62, 72, 16 · 371, 756, 432, 64 … Σ 1, 13, 150, 1623, 16857, 170208, …
  (−∞, 0]nonnegative
• A028338 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 8, 24, 9 · 48, 176, 108, 16 … Σ 1, 6, 41, 348, 3589, 43802, 618869, …
  (−∞, 0]nonnegative
• A038305 → basis A013611 → D
  312, 288 · 6084, 11232, 5184 … Σ 600, 22500, 750000, 23437500, …
  (−∞, 0]nonnegativerational roots
• A144436 → (θ+1)²
  1 · 1, 4 · 1, 32, 9 · 1, 92, 207, 16 … Σ 1, 5, 42, 316, 2636, 23880, 236064, …
  (−∞, 0]nonnegative
• A106800 → D
  1 · 3, 2 · 6, 14, 3 · 10, 50, 45, 4 … Σ 0, 0, 1, 5, 23, 109, 544, 2876, …
  (−∞, 0]nonnegative
• A094638 → basis A013611 → reverse
  1 · 4, 2 · 32, 28, 6 · 384, 464, 184, 24 … Σ 1, 6, 66, 1056, 22176, 576576, …
  (−∞, 0]nonnegativerational roots
• A121574 → basis A013609 → x ↦ x+1
  1 · 5, 2 · 28, 22, 4 · 155, 182, 68, 8 … Σ 1, 7, 54, 413, 3161, 24192, 185149, …
  (−∞, −1]nonnegative
• A207824 → basis A013609 → (θ+1)²
  1 · 6, 8 · 35, 96, 36 · 204, 848, 648, 128 … Σ 1, 14, 167, 1828, 18953, 189226, …
  (−∞, 0]nonnegative
• A013618 → basis A013610 → D+1
  936, 2970, 1089 · 15984, 92664, 147015, 35937 … Σ 4995, 291600, 16129125, 861131250, …
  (−∞, 0]nonnegativerational roots
• A126351 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 5 · 10, 33, 27 · 37, 182, 296, 159 … Σ 1, 8, 70, 674, 7138, 82730, 1042066, …
  [−1, 0]nonnegative
• A145597: Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.
  1 · 3, 3 · 6, 15, 6 · 10, 45, 45, 10 … Σ 1, 6, 27, 110, 429, 1638, 6188, 23256, …
  (−∞, 0)nonnegative
• A038329 → basis A013609 → basis A013610
  441, 756, 324 · 9261, 23814, 20412, 5832 … Σ 1521, 59319, 2313441, 90224199, …
  (−∞, 0]nonnegativerational roots
• A104684 → basis A013611 → D+1
  1 · 7, 4 · 45, 64, 16 · 291, 708, 432, 64 … Σ 1, 11, 125, 1495, 18305, 225947, …
  (−∞, 0]nonnegative