Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A144445 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 25, 25 · 1, 209, 621, 414 … Σ 1, 3, 51, 1245, 42151, 1843581, …
  [−1, 0]nonnegative
• A123519 → reverse → D+1
  10, 8, 1 · 28, 44, 15, 1 · 72, 176, 120, 24, 1 … Σ 19, 88, 393, 1708, 7275, 30512, 126417, …
  (−∞, 0]nonnegative
• A142963 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 12, 15 · 1, 33, 135, 111 … Σ 1, 4, 28, 280, 3616, 57184, 1069888, …
  [−1, 0]nonnegative
• A013609 → Veronese 2 → θ+1
  1 · 1 · 1, 8 · 1, 24 … Σ 1, 1, 9, 25, 97, 321, 1097, 3641, …
  (−∞, 0]nonnegative
• A038291 → basis A013609 → D
  2 · 40, 8 · 600, 240, 24 · 8000, 4800, 960, 64 … Σ 0, 2, 48, 864, 13824, 207360, 2985984, …
  (−∞, 0]nonnegativerational roots
• A082985 → reverse → D+1
  1 · 4, 1 · 10, 7, 1 · 21, 28, 10, 1 … Σ 1, 5, 18, 60, 191, 589, 1775, 5257, …
  (−∞, 0]nonnegative
• A038279 → basis A013609 → reverse
  1 · 2, 9 · 4, 36, 81 · 8, 108, 486, 729 … Σ 1, 11, 121, 1331, 14641, 161051, …
  (−∞, 0]nonnegativerational roots
• A304236 → (θ+1)²
  1 · 1 · 1, 12 · 1, 24 … Σ 1, 1, 13, 25, 118, 292, 979, 2611, …
  (−∞, 0]nonnegative
• A114687 → D
  2 · 6, 8 · 12, 48, 24 · 20, 160, 240, 64 … Σ 0, 2, 14, 84, 484, 2750, 15546, 87752, …
  (−∞, 0]nonnegative
• A356546 → x ↦ x+1 → reverse
  1 · 2, 4 · 6, 24, 24 · 20, 120, 240, 160 … Σ 1, 6, 54, 540, 5670, 61236, 673596, …
  (−∞, 0]nonnegativerational roots
• A038327 → basis A013610 → D
  3 · 78, 18 · 1521, 702, 81 · 26364, 18252, 4212, 324 … Σ 0, 3, 96, 2304, 49152, 983040, …
  (−∞, 0]nonnegativerational roots
• A008292 → γ (scale 1/2) (shift +1) → (θ+1)²
  1 · 1 · 1, 4 · 1, 16 … Σ 1, 1, 5, 17, 81, 411, 2393, 15329, …
  (−∞, 0]nonnegative
• A038243 → basis A013611 → x ↦ x/(x+1)
  1 · 6, 10 · 36, 120, 100 · 216, 1080, 1800, 1000 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  [−1, 0]nonnegativerational roots
• A286785 → D
  14, 4 · 72, 54, 6 · 330, 440, 132, 8 … Σ 18, 132, 910, 6120, 40698, 269192, …
  (−∞, 0]nonnegative
• A056939 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 10, 12 · 1, 23, 73, 55 … Σ 1, 4, 23, 152, 1101, 8508, 69007, …
  [−1, 0]nonnegative
• A272867 → basis A013609 → x ↦ x/(x+1)
  1 · 6, 24, 22 · 36, 288, 840, 1056, 484 … Σ 1, 52, 2704, 140608, 7311616, …
  [−1, 0]nonnegative
• A114655 → x ↦ x+1
  2 · 6, 2 · 22, 16, 2 · 90, 102, 30, 2 … Σ 2, 8, 40, 224, 1344, 8448, 54912, …
  (−∞, −1]nonnegative
• A062264 → reverse → θ+1
  1 · 5, 2 · 15, 24, 3 · 35, 126, 63, 4 … Σ 1, 7, 42, 228, 1155, 5577, 26026, …
  (−∞, 0]nonnegative
• A133080 → θ+1 → D
  2 · 0, 6 · 0, 6, 12 · 0, 0, 0, 20 … Σ 0, 2, 6, 18, 20, 50, 42, 98, …
  (−∞, 0]nonnegativerational roots
• A156992 → basis A013609 → θ+1
  1 · 4, 8 · 24, 96, 72 · 192, 1152, 1728, 768 … Σ 1, 12, 192, 3840, 92160, 2580480, …
  (−∞, 0]nonnegativerational roots
• A086645 → Veronese 2 → θ+1
  1 · 1 · 1, 2 · 1, 30 … Σ 1, 1, 3, 31, 144, 556, 2480, 11376, …
  (−∞, 0]nonnegative
• A013622 → x ↦ x/(x+1) → x ↦ x/(x+1)
  1 · 3, 11 · 9, 66, 121 · 27, 297, 1089, 1331 … Σ 1, 14, 196, 2744, 38416, 537824, …
  [−1, 0]nonnegativerational roots
• A039756 → reverse → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 26, 27, 4 … Σ 1, 3, 12, 58, 324, 2044, 14320, 109960, …
  (−∞, 0]nonnegative
• Veronese parent of A098172 → basis A007318
  1, 2, 1 · 2, 9, 16, 14, 6, 1 … Σ 4, 48, 576, 6912, 82944, 995328, …
  (−∞, −1]nonnegativerational roots
• A086645 → ⊙ C(n+1,k) (shift +2) → x ↦ x/(x+1)
  1, 14, 14 · 1, 48, 138, 92 · 1, 116, 762, 1292, 646 … Σ 29, 279, 2817, 29283, 309869, 3321447, …
  [−1, 0]nonnegative
• A008292 → γ (scale 1/4) → D
  2 · 16 · 88, 32 · 416, 544 … Σ 0, 0, 2, 16, 120, 960, 8400, 80640, …
  (−∞, 0]nonnegative
• narayana → basis A013609 → D
  2 · 10, 8 · 42, 72, 24 · 168, 448, 336, 64 … Σ 0, 2, 18, 138, 1016, 7380, 53358, …
  (−∞, 0]nonnegative
• A038338 → basis A013610 → (θ+1)²
  576, 6912, 11664 · 13824, 248832, 839808, 746496 … Σ 19152, 1848960, 162259200, 13374720000, …
  (−∞, 0]nonnegative
• A113953 → basis A007318 → D
  1 · 4, 2 · 11, 14, 3 · 30, 56, 30, 4 … Σ 0, 1, 6, 28, 120, 480, 1856, 6976, …
  (−∞, 0]nonnegative
• A038329 → basis A013611 → x ↦ x+1
  729, 648, 144 · 19683, 26244, 11664, 1728 … Σ 1521, 59319, 2313441, 90224199, …
  (−∞, −1]nonnegativerational roots
• A038207 → (θ+1)²
  1 · 2, 4 · 4, 16, 9 · 8, 48, 54, 16 … Σ 1, 6, 29, 126, 513, 1998, 7533, 27702, …
  (−∞, 0]nonnegative
• A166344 → x ↦ x+1
  8, 8, 1 · 32, 48, 18, 1 · 160, 320, 198, 38, 1 … Σ 17, 99, 717, 6219, 62877, 726219, …
  (−∞, −1]nonnegative
• A094638 → basis A013610 → θ+1
  1 · 2, 6 · 6, 42, 54 · 24, 276, 783, 648 … Σ 1, 8, 102, 1731, 36447, 914562, …
  (−∞, 0]nonnegative
• A027467 → basis A013611 → basis A013610
  1 · 20, 12 · 400, 480, 144 · 8000, 14400, 8640, 1728 … Σ 1, 32, 1024, 32768, 1048576, 33554432, …
  (−∞, 0]nonnegativerational roots
• A114687 → (θ+1)²
  1 · 1, 8 · 1, 24, 36 · 1, 48, 216, 128 … Σ 1, 9, 61, 393, 2481, 15473, 95677, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → (θ+1)² → D
  0, 18 · 0, 0, 48 · 0, 0, 0, 100 · 0, 0, 0, 0, 180 … Σ 18, 48, 100, 180, 294, 448, 648, 900, …
  (−∞, 0]nonnegativerational roots
• A049444: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 1.
  1 · -2, 1 · 6, -5, 1 · -24, 26, -9, 1 … Σ 1, -1, 2, -6, 24, -120, 720, -5040, …
  [3/2, ∞)rational roots
• A143683 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 12, 12 · 1, 22, 60, 40 … Σ 1, 3, 25, 123, 769, 4275, 25129, …
  [−1, 0]nonnegative
• A052553 → D+1
  1 · 1 · 2, 1 · 3, 2 … Σ 1, 1, 3, 5, 10, 18, 33, 59, …
  (−∞, 0]nonnegative
• A011973 → basis A013611 → D
  4 · 8 · 20, 32 · 40, 96 … Σ 0, 0, 4, 8, 52, 136, 560, 1624, …
  (−∞, 0]nonnegative
• A201637 → reverse (shift +1) → x ↦ x+1
  1 · 1, 1 · 3, 4, 1 · 15, 25, 11, 1 … Σ 1, 2, 8, 52, 472, 5504, 78416, 1320064, …
  (−∞, −1]nonnegative
• A038338 → basis A013610 → D+1
  2304, 4320, 1296 · 76032, 248832, 233280, 46656 … Σ 7920, 604800, 44064000, 3110400000, …
  (−∞, 0]nonnegativerational roots
• A225469 → (θ+1)²
  1 · 3, 4 · 9, 40, 9 · 27, 316, 189, 16 … Σ 1, 7, 58, 548, 5792, 67728, 868000, …
  (−∞, 0]nonnegative
• A081577 → basis A013609 → θ+1
  1 · 2, 4 · 6, 24, 12 · 16, 96, 120, 32 … Σ 1, 6, 42, 264, 1620, 9672, 56712, …
  (−∞, 0]nonnegative
• A286718: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.
  1 · 1, 1 · 4, 5, 1 · 28, 39, 12, 1 … Σ 1, 2, 10, 80, 880, 12320, 209440, …
  (−∞, −1]nonnegativerational roots
• A109692 → basis A007318
  1 · 2, 1 · 8, 10, 3 · 48, 100, 68, 15 … Σ 1, 3, 21, 231, 3465, 65835, 1514205, …
  (−∞, −1]nonnegativerational roots
• A028338 → basis A013609 → (θ+1)²
  1 · 2, 8 · 8, 48, 36 · 48, 352, 432, 128 … Σ 1, 10, 92, 960, 11584, 160384, 2517248, …
  (−∞, 0]nonnegative
• A132393 → basis A013610 → reverse
  1 · 3, 1 · 9, 9, 2 · 27, 54, 33, 6 … Σ 1, 4, 20, 120, 840, 6720, 60480, …
  (−∞, 0]nonnegativerational roots
• A008292 → γ (shift +1) → x ↦ x+1 → D
  2 · 8 · 54, 32 · 324, 272 … Σ 0, 0, 2, 8, 86, 596, 6258, 60144, …
  (−∞, 0]nonnegative
• A056242 → D+1
  1 · 3, 2 · 6, 13, 4 · 10, 41, 40, 8 … Σ 1, 5, 23, 99, 408, 1632, 6388, 24596, …
  (−∞, 0]nonnegative