Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A081578 → basis A013610 → reverse
  1 · 3, 2 · 9, 21, 7 · 27, 108, 90, 20 … Σ 1, 5, 37, 245, 1669, 11285, 76453, …
  (−∞, 0]nonnegative
• A269944 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 4, 13, 10 … Σ 1, 1, 3, 27, 513, 16929, 863379, …
  [−1, 0]nonnegativerational roots
• A078812 → basis A013611 → D
  4 · 24, 32 · 100, 288, 192 · 360, 1632, 2304, 1024 … Σ 0, 4, 56, 580, 5320, 45680, 376264, …
  (−∞, 0]nonnegative
• A034870 → basis A013610 → θ+1
  1 · 4, 24, 27 · 16, 192, 648, 864, 405 … Σ 1, 55, 2125, 71875, 2265625, 68359375, …
  (−∞, 0]nonnegativerational roots
• A178820 → Veronese 2 (j=1) → D
  20 · 140 · 560, 112 · 1680, 1008 … Σ 0, 0, 0, 20, 140, 672, 2688, 9600, …
  (−∞, 0]nonnegative
• A274496 → D → θ+1
  2 · 4 · 8, 16 · 16, 32 … Σ 0, 0, 2, 4, 24, 48, 192, 384, …
  (−∞, 0]nonnegativerational roots
• A145324 → basis A013609 → (θ+1)²
  1 · 3, 16 · 12, 136, 216 · 60, 1064, 3528, 3072 … Σ 1, 19, 364, 7724, 183404, 4851620, …
  (−∞, 0]nonnegative
• A154986 → γ → D
  2 · 8 · 20, 48 · 40, 368 … Σ 0, 0, 2, 8, 68, 408, 3798, 30384, …
  (−∞, 0]nonnegative
• A099040 → basis A013611 → D
  8 · 40, 128 · 160, 1024, 1536 … Σ 0, 8, 168, 2720, 39360, 535200, …
  (−∞, 0]nonnegative
• A036969 → D+1
  1 · 2, 1 · 6, 7, 1 · 22, 49, 17, 1 … Σ 1, 3, 14, 89, 737, 7696, 98093, …
  (−∞, 0]nonnegative
• A039756 → reverse → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 52, 81, 16 … Σ 1, 5, 26, 150, 964, 6836, 52936, …
  (−∞, 0]nonnegative
• A099040 → basis A013610 → D
  6 · 30, 72 · 120, 576, 648 · 432, 3096, 7128, 5184 … Σ 0, 6, 102, 1344, 15840, 175488, …
  (−∞, 0]nonnegative
• A358735 → basis A007318 → x ↦ x/(x+1)
  1 · 2, 3 · 7, 22, 17 · 33, 159, 253, 133 … Σ 1, 5, 46, 578, 9064, 169904, 3701632, …
  [−1, −1/2)nonnegative
• A130534 → Veronese 2 (j=1) → D+1
  1 · 3 · 12, 1 · 60, 10 … Σ 0, 1, 3, 13, 70, 447, 3297, 27576, …
  (−∞, 0]nonnegative
• A132774 → Veronese 2 (j=1) → (θ+1)²
  3 · 4 · 0, 28 · 0, 32 … Σ 0, 3, 4, 28, 32, 99, 108, 240, …
  (−∞, 0]nonnegativerational roots
• A257615: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.
  1 · 5, 5 · 25, 70, 25 · 125, 715, 715, 125 … Σ 1, 10, 120, 1680, 26880, 483840, …
  (−∞, 0)nonnegative
• A008459 → D → θ+1
  1 · 4, 4 · 9, 36, 9 · 16, 144, 144, 16 … Σ 0, 1, 8, 54, 320, 1750, 9072, 45276, …
  (−∞, 0]nonnegative
• A127648 → Veronese 2 (j=1) → D
  4 · 0, 12 · 0, 0, 24 · 0, 0, 0, 40 … Σ 0, 0, 0, 0, 4, 0, 12, 0, …
  (−∞, 0]nonnegativerational roots
• A104684 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 20, 90, 132, 63 … Σ 1, 5, 37, 305, 2641, 23525, 213445, …
  [−1, 0]nonnegative
• A131689 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 3 · 0, 1, 8, 13 … Σ 1, 1, 4, 22, 160, 1456, 15904, 202672, …
  [−1, 0]nonnegative
• A113214 → x ↦ x−5
  1 · -3, 1 · 10, -7, 1 · -35, 37, -11, 1 … Σ 1, -2, 4, -8, 16, -32, 64, -128, …
  (1, 5]
• A039755 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 24, 106, 152, 71 … Σ 1, 5, 37, 353, 4009, 52253, 764557, …
  [−1, 0]nonnegative
• A028421: theta_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers …
  1 · 1, 2 · 2, 6, 3 · 6, 22, 18, 4 … Σ 1, 3, 11, 50, 274, 1764, 13068, 109584, …
  (−∞, 0)nonnegative
• A078812 → D+1
  1 · 3, 1 · 7, 6, 1 · 14, 22, 9, 1 … Σ 1, 4, 14, 46, 145, 444, 1331, 3926, …
  (−∞, 0]nonnegative
• A135838 → x ↦ x+1
  1 · 4, 2 · 8, 8, 2 · 32, 48, 24, 4 … Σ 1, 6, 18, 108, 324, 1944, 5832, 34992, …
  (−∞, −1]nonnegativerational roots
• A099097 → basis A007318 → D+1
  1 · 6, 3 · 29, 37, 9 · 126, 267, 168, 27 … Σ 1, 9, 75, 588, 4436, 32580, 234596, …
  (−∞, 0]nonnegative
• A062196 → reverse
  1 · 3, 1 · 6, 8, 1 · 10, 30, 15, 1 … Σ 1, 4, 15, 56, 210, 792, 3003, 11440, …
  (−∞, 0]nonnegative
• A127648 → D → D
  6 · 0, 24 · 0, 0, 60 · 0, 0, 0, 120 … Σ 0, 0, 0, 6, 24, 60, 120, 210, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A030528, shifted → basis A013610 → reverse
  1 · -3, -1 · 9, 6, 1 · -27, -27, -9, -1 … Σ 1, -4, 16, -64, 256, -1024, 4096, …
  (−∞, −1]rational roots
• A109983 → θ+1
  1 · 0, 2, 6 · 0, 0, 3, 24, 30 … Σ 1, 8, 57, 384, 2505, 16008, 100849, …
  (−∞, 0]nonnegative
• A104684 → basis A013611 → (θ+1)²
  1 · 3, 16 · 13, 128, 144 · 63, 912, 2160, 1024 … Σ 1, 19, 285, 4159, 59905, 849507, …
  (−∞, 0]nonnegative
• A121574 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 5 · 10, 34, 28 · 33, 169, 283, 155 … Σ 1, 8, 72, 640, 5696, 50688, 451072, …
  [−1, 0]nonnegative
• A038319 → basis A013610 → D+1
  736, 930, 225 · 15616, 33120, 20925, 3375 … Σ 1891, 73036, 2710981, 97893226, …
  (−∞, 0]nonnegativerational roots
• A225471 → basis A007318 → D+1
  1 · 5, 1 · 44, 14, 1 · 560, 224, 27, 1 … Σ 1, 6, 59, 812, 14389, 312114, 8011695, …
  (−∞, 0]nonnegative
• A253284 → x ↦ x/(x+1)
  1 · 2, 4 · 6, 30, 36 · 24, 216, 600, 528 … Σ 1, 6, 72, 1368, 36000, 1213920, …
  [−1, 0]nonnegative
• A257616 → D
  2 · 32, 8 · 312, 624, 24 · 2656, 17472, 7968, 64 … Σ 0, 2, 40, 960, 28160, 985600, 40212480, …
  (−∞, 0]nonnegative
• A113214 → θ+1
  1 · 2, 2 · 0, 6, 3 · 0, 4, 12, 4 … Σ 1, 4, 9, 20, 40, 78, 147, 272, …
  (−∞, 0]nonnegative
• A304236 → basis A007318 → basis A013609
  1 · 1 · 7, 6 · 13, 12 … Σ 1, 1, 13, 25, 181, 481, 2653, 8425, …
  (−∞, −1]nonnegative
• A028246 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 6 · 1, 10, 29, 26 … Σ 1, 3, 12, 66, 480, 4368, 47712, 608016, …
  [−1, 0)nonnegative
• A152842 → reverse → x ↦ x/(x+1)
  1 · 3, 4 · 3, 10, 8 · 9, 42, 64, 32 … Σ 1, 7, 21, 147, 441, 3087, 9261, 64827, …
  [−1, 0]nonnegativerational roots
• A200536 → basis A013609 → D
  14, 16 · 168, 584, 672, 256 … Σ 0, 30, 1680, 70560, 2634240, 92198400, …
  (−∞, 0]nonnegativerational roots
• A122542 → reverse → θ+1
  1 · 1 · 1, 4 · 1, 8, 6 … Σ 1, 1, 5, 15, 45, 129, 361, 991, …
  (−∞, 0]nonnegative
• A038220 → basis A007318
  1 · 5, 2 · 25, 20, 4 · 125, 150, 60, 8 … Σ 1, 7, 49, 343, 2401, 16807, 117649, …
  (−∞, −1]nonnegativerational roots
• A038255 → basis A013610 → D
  3 · 42, 18 · 441, 378, 81 · 4116, 5292, 2268, 324 … Σ 0, 3, 60, 900, 12000, 150000, 1800000, …
  (−∞, 0]nonnegativerational roots
• A110441 → basis A013609 → D
  2 · 16, 8 · 88, 96, 24 · 416, 720, 384, 64 … Σ 0, 2, 24, 208, 1584, 11256, 76608, …
  (−∞, 0]nonnegative
• A094638 → Veronese 2 (j=1) → x ↦ x+1
  1 · 3 · 12, 6 · 60, 50 … Σ 0, 1, 3, 18, 110, 945, 8547, 96796, …
  (−∞, −1]nonnegative
• A137320 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 8 · 6, 42, 72 · 24, 228, 708, 720 … Σ 1, 10, 120, 1680, 26880, 483840, …
  [−1, 0]nonnegativerational roots
• A054142 → Veronese 2 (j=1) → x ↦ x+1
  1 · 3 · 6, 1 · 17, 10 … Σ 0, 1, 3, 7, 27, 83, 263, 855, …
  (−∞, −1]nonnegative
• A049374 → D+1
  1 · 7, 1 · 60, 20, 1 · 612, 348, 39, 1 … Σ 1, 8, 81, 1000, 14549, 242784, 4554613, …
  (−∞, 0]nonnegative
• A257616 → D+1
  1 · 4, 2 · 36, 40, 4 · 320, 936, 336, 8 … Σ 1, 6, 80, 1600, 42240, 1379840, …
  (−∞, 0]nonnegative