Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A135065 → (θ+1)²
  1 · 4, 16 · 9, 72, 81 · 16, 192, 432, 256 … Σ 1, 20, 162, 896, 4000, 15552, 54880, …
  (−∞, 0]nonnegative
• A143491 → basis A013610 → (θ+1)²
  12, 84, 81 · 60, 564, 972, 432 … Σ 177, 2028, 23904, 295956, 3879108, …
  (−∞, 0]nonnegative
• A038284 → basis A007318 → x ↦ x+1
  400, 240, 36 · 8000, 7200, 2160, 216 … Σ 676, 17576, 456976, 11881376, …
  (−∞, −1]nonnegativerational roots
• A026374: Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) +…
  1 · 1, 1 · 1, 3, 1 · 1, 4, 4, 1 … Σ 1, 2, 5, 10, 25, 50, 125, 250, …
  (−∞, 0]nonnegative
• A225476 → x ↦ x+1
  1 · 2, 1 · 7, 8, 2 · 38, 67, 36, 6 … Σ 1, 3, 17, 147, 1697, 24483, 423857, …
  (−3, −1)nonnegative
• A089627 → x ↦ x+1 → x ↦ x+1
  1 · 1 · 5, 2 · 13, 6 … Σ 1, 1, 7, 19, 91, 331, 1441, 5797, …
  (−∞, −1]nonnegative
• A372873 → D+1
  1 · 2, 2 · 1, 9, 4 · 0, 12, 30, 8 … Σ 1, 4, 14, 50, 179, 632, 2192, 7478, …
  (−∞, 0]nonnegative
• A282629 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 17, 25 · 1, 66, 237, 199 … Σ 1, 5, 43, 503, 7105, 116453, 2164123, …
  [−1, 0]nonnegative
• A091042 → D+1
  1 · 4, 3 · 11, 20, 5 · 22, 91, 56, 7 … Σ 1, 7, 36, 176, 832, 3840, 17408, 77824, …
  (−∞, 0]nonnegative
• A008556 → (θ+1)²
  1 · 2 · 6, 8 · 20, 48 … Σ 1, 2, 14, 68, 364, 1912, 10064, 52768, …
  (−∞, 0]nonnegative
• A331333 → D
  2 · 8, 8 · 36, 72, 24 · 192, 576, 384, 64 … Σ 0, 2, 16, 132, 1216, 12560, 144384, …
  (−∞, 0]nonnegative
• A160562 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 128, 164, 38, 1 … Σ 1, 3, 25, 331, 6193, 157971, 5155657, …
  (−∞, −1]nonnegative
• A113413 → basis A013611 → x ↦ x/(x+1)
  1 · 3, 7 · 7, 38, 47 · 17, 143, 379, 317 … Σ 1, 10, 92, 856, 7952, 73888, 686528, …
  [−1, 0]nonnegative
• A099093 → reverse → x ↦ x+1
  1 · 3 · 12, 3 · 45, 18 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, −1]nonnegative
• A038328 → basis A013609 → reverse
  16, 112, 196 · 64, 672, 2352, 2744 … Σ 324, 5832, 104976, 1889568, 34012224, …
  (−∞, 0]nonnegativerational roots
• A156992 → basis A013609 → D
  4 · 48, 48 · 576, 1152, 576 … Σ 0, 4, 96, 2304, 61440, 1843200, …
  (−∞, 0]nonnegativerational roots
• A130534 → basis A008277 → θ+1
  1 · 2, 2 · 6, 12, 3 · 24, 72, 36, 4 … Σ 1, 4, 21, 136, 1045, 9276, 93289, …
  (−∞, 0]nonnegative
• A216973 → θ+1 → (θ+1)²
  1 · 2, 16 · 3, 48, 81 · 4, 96, 324, 256 … Σ 0, 1, 18, 132, 680, 2880, 10752, 36736, …
  (−∞, 0]nonnegative
• A188440 → basis A007318
  1 · 1 · 3, 2 · 3, 2 … Σ 1, 1, 5, 5, 25, 25, 125, 125, …
  (−∞, −1]nonnegativerational roots
• A038327 → basis A013609 → (θ+1)²
  1 · 13, 8 · 169, 208, 36 · 2197, 4056, 1404, 128 … Σ 1, 21, 413, 7785, 142425, 2548125, …
  (−∞, 0]nonnegative
• A120910 → basis A013609 → θ+1
  3 · 9, 12 · 27, 72, 36 · 81, 324, 324, 96 … Σ 3, 21, 135, 825, 4875, 28125, 159375, …
  (−∞, 0]nonnegativerational roots
• A116412 → basis A013611 → reverse
  1 · 4, 4 · 16, 32, 13 · 64, 192, 168, 43 … Σ 1, 8, 61, 467, 3574, 27353, 209341, …
  (−∞, 0]nonnegative
• A367270 → D
  2 · 6, 6 · 12, 36, 12 · 20, 120, 120, 20 … Σ 0, 0, 2, 12, 60, 280, 1260, 5544, …
  (−∞, 0]nonnegative
• A290315 → reverse
  1 · 2, 1 · 12, 16, 1 · 120, 284, 66, 1 … Σ 1, 3, 29, 471, 10729, 314427, 11265701, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−3
  1 · -9, 3 · 72, -51, 9 · -567, 621, -225, 27 … Σ 1, -6, 30, -144, 684, -3240, 15336, …
  (5/3, 3]
• A120910 → θ+1
  3 · 6, 6 · 12, 24, 9 · 24, 72, 54, 12 … Σ 3, 12, 45, 162, 567, 1944, 6561, 21870, …
  (−∞, 0]nonnegativerational roots
• A152198 → basis A013609 → D+1
  1 · 1 · 4, 2 · 4, 2 … Σ 1, 1, 6, 6, 32, 32, 160, 160, …
  (−∞, 0]nonnegativerational roots
• A023531 → Veronese 2 → basis A007318
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … Σ 1, 0, 2, 0, 4, 0, 8, 0, …
  (−∞, −1]nonnegativerational roots
• A128100 → basis A008277
  1 · 1 · 3, 1 · 5, 2 … Σ 1, 1, 4, 7, 20, 43, 113, 267, …
  (−∞, 0]nonnegative
• A089627 → x ↦ x+1 → D
  2 · 6 · 24, 12 · 80, 60 … Σ 0, 0, 2, 6, 36, 140, 630, 2562, …
  (−∞, 0]nonnegative
• A038329 → basis A013609
  225, 180, 36 · 3375, 4050, 1620, 216 … Σ 441, 9261, 194481, 4084101, 85766121, …
  (−∞, 0]nonnegativerational roots
• A159854 → basis A013610 → reverse
  9, 6, 1 · 27, 36, 15, 2 · 81, 162, 117, 36, 4 … Σ 16, 80, 400, 2000, 10000, 50000, …
  (−∞, 0]nonnegativerational roots
• A104684 → D → D
  2 · 24, 6 · 180, 120, 12 · 1120, 1260, 360, 20 … Σ 0, 0, 2, 30, 312, 2760, 22290, 169806, …
  (−∞, 0]nonnegative
• A172431 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 9 · 8, 46, 65 · 21, 177, 489, 441 … Σ 1, 12, 119, 1128, 10561, 98532, 918359, …
  [−1, 0]nonnegative
• A274496 → basis A013611 → x ↦ x+1
  1 · 2 · 12, 8 · 24, 16 … Σ 1, 2, 20, 40, 400, 800, 8000, 16000, …
  (−∞, −1]nonnegativerational roots
• A201701 → D
  1 · 3 · 8, 2 · 20, 10 … Σ 0, 0, 1, 3, 10, 30, 87, 245, …
  (−∞, 0]nonnegative
• A113413 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 7, 32, 34 · 17, 120, 270, 194 … Σ 1, 9, 73, 601, 4937, 40569, 333353, …
  [−1, 0]nonnegative
• A128099 → D → D
  8 · 24 · 48, 48 · 80, 192 … Σ 0, 0, 0, 0, 8, 24, 96, 272, …
  (−∞, 0]nonnegative
• A081577 → basis A013610 → θ+1
  1 · 2, 6 · 6, 36, 27 · 16, 144, 270, 108 … Σ 1, 8, 69, 538, 4055, 29676, 212809, …
  (−∞, 0]nonnegative
• A225471 → reverse
  1 · 1, 3 · 1, 10, 21 · 1, 21, 131, 231 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
  (−∞, 0]nonnegativerational roots
• A113214 → x ↦ x−4
  1 · -2, 1 · 4, -5, 1 · -8, 18, -8, 1 … Σ 1, -1, 0, 3, -9, 18, -27, 27, …
  (0, 4]
• A132393 → basis A008277 → (θ+1)²
  1 · 1, 4 · 2, 16, 9 · 6, 72, 81, 16 … Σ 1, 5, 27, 175, 1337, 11781, 117595, …
  (−∞, 0]nonnegative
• A188440 → basis A007318 → D+1
  1 · 1 · 5, 2 · 5, 2 … Σ 1, 1, 7, 7, 45, 45, 275, 275, …
  (−∞, 0]nonnegativerational roots
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → D
  2 · 14, 6 · 66, 60, 12 · 256, 360, 156, 20 … Σ 0, 2, 20, 138, 792, 4050, 19116, 85050, …
  (−∞, 0]nonnegative
• A038208 → basis A013610
  1 · 4, 6 · 16, 48, 36 · 64, 288, 432, 216 … Σ 1, 10, 100, 1000, 10000, 100000, …
  [−1, 0]nonnegativerational roots
• delannoy → x ↦ x+1 → basis A007318
  1 · 3, 1 · 11, 7, 1 · 39, 37, 11, 1 … Σ 1, 4, 19, 88, 409, 1900, 8827, 41008, …
  (−∞, −1]nonnegative
• A269939 → reverse
  1 · 1 · 3, 1 · 15, 10, 1 … Σ 1, 1, 4, 26, 236, 2752, 39208, 660032, …
  (−∞, 0]nonnegative
• A207823 → θ+1
  1 · 4, 2 · 15, 16, 3 · 56, 92, 36, 4 … Σ 1, 6, 34, 188, 1021, 5468, 28959, …
  (−∞, 0]nonnegative
• A038330 → D+1
  240, 128, 16 · 3456, 2880, 768, 64 … Σ 384, 7168, 131072, 2359296, 41943040, …
  (−∞, 0]nonnegativerational roots
• A038324 → x ↦ x/(x+1)
  1 · 11, 21 · 121, 462, 441 · 1331, 7623, 14553, 9261 … Σ 1, 32, 1024, 32768, 1048576, 33554432, …
  [−1, 0]nonnegativerational roots