Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A173018 → x ↦ x+1 → D+1
  1 · 1 · 3, 1 · 12, 8, 1 … Σ 1, 1, 4, 21, 142, 1175, 11476, 129073, …
  (−∞, 0]nonnegative
• A038303 → basis A013611 → θ+1
  1 · 11, 8 · 121, 176, 48 · 1331, 2904, 1584, 256 … Σ 1, 19, 345, 6075, 104625, 1771875, …
  (−∞, 0]nonnegativerational roots
• A038329 → basis A013609 → D
  180, 72 · 4050, 3240, 648 · 81000, 97200, 38880, 5184 … Σ 252, 7938, 222264, 5834430, 147027636, …
  (−∞, 0]nonnegativerational roots
• A208509 → D+1
  1 · 3 · 6, 1 · 12, 5 … Σ 1, 3, 7, 17, 40, 92, 208, 464, …
  (−∞, 0]nonnegative
• A127648 → D
  2 · 0, 6 · 0, 0, 12 · 0, 0, 0, 20 … Σ 0, 0, 2, 6, 12, 20, 30, 42, …
  (−∞, 0]nonnegativerational roots
• A372873 → D
  2 · 1, 8 · 0, 12, 24 · 0, 2, 72, 64 … Σ 0, 2, 9, 36, 138, 510, 1827, 6384, …
  (−∞, 0]nonnegative
• A102547 → reverse → D+1
  1, 1 · 0, 2, 1 · 0, 0, 3, 1 · 0, 0, 3, 5, 1 … Σ 2, 3, 4, 9, 16, 25, 42, 70, …
  (−∞, 0]nonnegative
• A008459 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 22, 20, 4 · 136, 180, 72, 8 … Σ 1, 6, 46, 396, 3606, 33876, 324556, …
  (−∞, −1]nonnegative
• A075497 → θ+1
  1 · 2, 2 · 4, 12, 3 · 8, 56, 36, 4 … Σ 1, 4, 19, 104, 641, 4380, 32803, …
  (−∞, 0]nonnegative
• A008292 → γ → D+1
  1 · 1 · 3, 2 · 9, 8 … Σ 1, 1, 5, 17, 93, 513, 3477, 25569, …
  (−∞, 0]nonnegative
• A114655 → x ↦ x+1 → θ+1
  2 · 6, 4 · 22, 32, 6 · 90, 204, 90, 8 … Σ 2, 10, 60, 392, 2688, 19008, 137280, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013611 → reverse
  1 · 16, 8, 1 · 256, 256, 96, 16, 1 … Σ 1, 25, 625, 15625, 390625, 9765625, …
  (−∞, 0]nonnegativerational roots
• A038325 → θ+1
  1 · 11, 22 · 121, 484, 363 · 1331, 7986, 11979, 5324 … Σ 1, 33, 968, 26620, 702768, 18037712, …
  (−∞, 0]nonnegativerational roots
• A038333 → basis A013611 → reverse
  784, 1064, 361 · 21952, 44688, 30324, 6859 … Σ 2209, 103823, 4879681, 229345007, …
  (−∞, 0]nonnegativerational roots
• A086645 → γ → θ+1
  1 · 1 · 1, 8 · 1, 24 … Σ 1, 1, 9, 25, 97, 321, 1097, 3641, …
  (−∞, 0]nonnegative
• A075501 → D
  18, 2 · 252, 72, 3 · 3240, 1800, 180, 4 … Σ 20, 327, 5224, 86441, 1517676, …
  (−∞, 0]nonnegative
• A034870 → basis A013610 → (θ+1)²
  1 · 4, 48, 81 · 16, 384, 1944, 3456, 2025 … Σ 1, 133, 7825, 353125, 13890625, …
  (−∞, 0]nonnegative
• A178820 → basis A013611 → reverse
  1 · 16, 8 · 160, 160, 40 · 1280, 1920, 960, 160 … Σ 1, 24, 360, 4320, 45360, 435456, …
  (−∞, −1]nonnegativerational roots
• A081580 → basis A007318 → reverse
  1 · 1, 2 · 1, 9, 9 · 1, 16, 42, 28 … Σ 1, 3, 19, 87, 451, 2223, 11179, 55767, …
  (−∞, 0]nonnegative
• A111125 → D
  1 · 5, 2 · 14, 14, 3 · 30, 54, 27, 4 … Σ 0, 1, 7, 31, 115, 390, 1254, 3893, …
  (−∞, 0]nonnegative
• A144445 → D
  1 · 23, 2 · 206, 412, 3 · 1677, 12682, 5031, 4 … Σ 0, 1, 25, 621, 19394, 734520, 32816325, …
  (−∞, 0]nonnegative
• A081579 → basis A013609 → (θ+1)²
  1 · 2, 8 · 8, 64, 36 · 24, 288, 504, 128 … Σ 1, 10, 108, 944, 7920, 62688, 480320, …
  (−∞, 0]nonnegative
• A257617 → x ↦ x+1
  1 · 4, 2 · 44, 44, 4 · 792, 1188, 412, 8 … Σ 1, 6, 92, 2400, 86696, 4003656, …
  (−∞, −1]nonnegative
• A027466 → basis A013609
  1 · 8, 2 · 64, 32, 4 · 512, 384, 96, 8 … Σ 1, 10, 100, 1000, 10000, 100000, …
  (−∞, −1]nonnegativerational roots
• A013612 → basis A013610 → D+1
  216, 630, 225 · 1836, 9720, 14175, 3375 … Σ 1071, 29106, 750141, 18670176, …
  (−∞, 0]nonnegativerational roots
• A130123 → basis A013609 → D
  4 · 16, 32 · 48, 192, 192 · 128, 768, 1536, 1024 … Σ 0, 4, 48, 432, 3456, 25920, 186624, …
  (−∞, 0]nonnegativerational roots
• A038322 → basis A013610 → reverse
  1 · 24, 19 · 576, 912, 361 … Σ 1, 43, 1849, 79507, 3418801, 147008443, …
  (−∞, 0]nonnegativerational roots
• A116666 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 8, 12 · 1, 12, 36, 32 … Σ 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A297191, shifted → basis A008288 → D+1
  1 · 4, 5, 1 · 8, 33, 34, 11, 1 … Σ 1, 10, 87, 676, 4925, 34446, 234227, …
  (−∞, 0]nonnegative
• A038319 → basis A013609 → x ↦ x+1
  676, 520, 100 · 17576, 20280, 7800, 1000 … Σ 1296, 46656, 1679616, 60466176, …
  (−∞, −1]nonnegativerational roots
• A036969 → D
  1 · 5, 2 · 21, 28, 3 · 85, 294, 90, 4 … Σ 0, 1, 7, 52, 473, 5263, 70180, 1103781, …
  (−∞, 0]nonnegative
• A091044 → basis A007318 → θ+1
  1 · 4, 4 · 16, 32, 9 · 64, 192, 120, 16 … Σ 1, 8, 57, 392, 2633, 17376, 113105, …
  (−∞, 0]nonnegative
• A059473 → basis A013610 → D
  6 · 54, 72 · 360, 1008, 648 · 2136, 9144, 12312, 5184 … Σ 0, 6, 126, 2016, 28776, 385632, …
  (−∞, 0]nonnegative
• A059473 → x ↦ x+1 → reverse
  1 · 2, 4 · 4, 18, 18 · 8, 56, 120, 80 … Σ 1, 6, 40, 264, 1744, 11520, 76096, …
  (−∞, 0]nonnegative
• A038305 → θ+1
  100, 120, 27 · 1000, 1800, 810, 108 … Σ 247, 3718, 54925, 799708, 11510083, …
  (−∞, 0]nonnegativerational roots
• A038221 → x ↦ x+1 → D
  3 · 36, 18 · 324, 324, 81 · 2592, 3888, 1944, 324 … Σ 0, 3, 54, 729, 8748, 98415, 1062882, …
  (−∞, 0]nonnegativerational roots
• A127648 → Veronese 2 (j=1) → D+1
  2 · 4, 4 · 0, 12, 6 · 0, 0, 24, 8 … Σ 0, 0, 2, 0, 8, 0, 18, 0, …
  (−∞, 0]nonnegativerational roots
• A038271 → basis A013611 → D
  20 · 480, 800 · 8640, 28800, 24000 … Σ 0, 20, 1280, 61440, 2621440, 104857600, …
  (−∞, 0]nonnegativerational roots
• A038305 → basis A013611 → θ+1
  169, 624, 432 · 2197, 12168, 16848, 6912 … Σ 1225, 38125, 1140625, 33203125, …
  (−∞, 0]nonnegativerational roots
• A027465 → Veronese 2 → D
  1 · 9 · 54, 2 · 270, 30 … Σ 0, 0, 1, 9, 56, 300, 1488, 7056, …
  (−∞, 0]nonnegative
• A048854 → reverse
  1 · 1, 2 · 1, 12, 12 · 1, 30, 180, 120 … Σ 1, 3, 25, 331, 5937, 133651, 3609673, …
  (−∞, 0]nonnegative
• A064861 → Veronese 2 (j=1) → x ↦ x+1
  3 · 9, 4 · 18, 12 · 54, 54, 8 … Σ 3, 13, 30, 116, 279, 1069, 2580, 9878, …
  (−∞, −1]nonnegative
• A062145 → reverse → x ↦ x+1
  1 · 5, 1 · 21, 12, 1 · 84, 84, 21, 1 … Σ 1, 6, 34, 190, 1059, 5908, 33028, …
  (−∞, −1]nonnegative
• A257616 → x ↦ x+1
  1 · 4, 2 · 40, 40, 4 · 640, 960, 336, 8 … Σ 1, 6, 84, 1944, 61776, 2496096, …
  (−∞, −1]nonnegative
• A123519 → basis A007318 → D+1
  25, 22, 4 · 117, 164, 68, 8 · 525, 1020, 684, 184, 16 … Σ 51, 357, 2429, 16187, 106175, 687785, …
  (−∞, 0]nonnegative
• A048994: Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.
  1 · 0, 1 · 0, -1, 1 · 0, 2, -3, 1 … Σ 1, 1, 0, 0, 0, 0, 0, 0, …
  [0, ∞)rational roots
• A122431 → (θ+1)²
  1, 12, 27, 16 · 0, 4, 27, 48, 25 … Σ 56, 104, 168, 248, 344, 456, 584, 728, …
  (−∞, 0]nonnegative
• A090582 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 24, 108, 158, 75 … Σ 1, 5, 37, 365, 4501, 66605, 1149877, …
  [−1, 0]nonnegative
• A098158 → D+1
  1 · 1, 1 · 1, 3, 1 · 0, 6, 6, 1 … Σ 1, 2, 5, 13, 32, 76, 176, 400, …
  (−∞, 0]nonnegative
• A085880 → basis A013609 → D
  2 · 16, 16 · 120, 240, 120 · 896, 2688, 2688, 896 … Σ 0, 2, 32, 480, 7168, 107520, 1622016, …
  (−∞, 0]nonnegativerational roots