Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• basis-map coefficients of A297191, shifted → (θ+1)² → D+1
  1 · 0, 18, 9 · 0, 0, 0, 100, 25 … Σ 1, 27, 125, 343, 729, 1331, 2197, 3375, …
  (−∞, 0]nonnegativerational roots
• A137338 → x ↦ x+1
  1 · 0, 1 · -2, -1, 1 · 4, -3, -3, 1 … Σ 1, 1, -2, -1, 10, -25, 40, -25, …
  (−∞, ∞)
• A086645 → θ+1
  1 · 1, 2 · 1, 12, 3 · 1, 30, 45, 4 … Σ 1, 3, 16, 80, 384, 1792, 8192, 36864, …
  (−∞, 0]nonnegative
• A136572 → basis A013611 → x ↦ x/(x+1)
  1 · 1, 5 · 2, 20, 50 · 6, 90, 450, 750 … Σ 1, 6, 72, 1296, 31104, 933120, …
  [−1, 0]nonnegativerational roots
• A056242 → x ↦ x+1 → D
  2 · 13, 8 · 65, 80, 24 · 292, 538, 324, 64 … Σ 0, 2, 21, 169, 1218, 8256, 53811, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A013610 → basis A007318
  16, 24, 9 · 64, 144, 108, 27 · 256, 768, 864, 432, 81 … Σ 49, 343, 2401, 16807, 117649, 823543, …
  (−∞, −1]nonnegativerational roots
• A099097 → basis A007318 → x ↦ x/(x+1)
  1 · 3, 6 · 10, 39, 38 · 33, 192, 372, 240 … Σ 1, 9, 87, 837, 8055, 77517, 745983, …
  [−1, 0]nonnegative
• A125092 → D+1 → x ↦ x+1
  1 · 9, 4 · 44, 44, 9 · 170, 264, 123, 16 … Σ 1, 13, 97, 573, 2961, 14013, 62289, …
  (−∞, −1]nonnegative
• A008277 → θ+2 (shift +1) → D+1
  2 · 5, 3 · 11, 17, 4 · 23, 69, 39, 5 … Σ 2, 8, 32, 136, 622, 3060, 16130, 90688, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → θ+1 → (θ+1)²
  1 · 0, -8 · 0, 0, 54 · 0, 0, 0, -384 … Σ 1, -8, 54, -384, 3000, -25920, 246960, …
  (−∞, 0]rational roots
• A081578 → x ↦ x+1 → D+1
  1 · 3, 1 · 14, 9, 1 · 50, 54, 15, 1 … Σ 1, 4, 24, 120, 612, 3024, 14796, 71496, …
  (−∞, 0]nonnegative
• A130534 → D → (θ+1)²
  1 · 3, 8 · 11, 48, 27 · 50, 280, 270, 64 … Σ 0, 1, 11, 86, 664, 5454, 48642, 472968, …
  (−∞, 0]nonnegative
• A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
  1 · 7, 11 · 49, 154, 121 · 343, 1617, 2541, 1331 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, 0)[−1, 0]nonnegative+1
• A038320 → basis A013611 → D+1
  1105, 1968, 576 · 25721, 79560, 70848, 13824 … Σ 3649, 189953, 9442177, 454947521, …
  (−∞, 0]nonnegativerational roots
• A013622 → basis A013610 → (θ+1)²
  1 · 8, 60 · 64, 960, 2025 · 512, 11520, 48600, 54000 … Σ 1, 68, 3049, 114632, 3898201, …
  (−∞, 0]nonnegative
• A152198 → basis A013609 → basis A007318
  1 · 1 · 4, 2 · 4, 2 … Σ 1, 1, 6, 6, 36, 36, 216, 216, …
  (−∞, −1]nonnegativerational roots
• A391533 → reverse → D
  4, 8 · 5, 16, 12 · 6, 26, 36, 16 · 7, 38, 75, 64, 20 … Σ 12, 33, 84, 204, 480, 1104, 2496, 5568, …
  (−∞, 0]nonnegativerational roots
• A159854 → basis A013610 → x ↦ x+1
  16, 24, 9 · 80, 168, 117, 27 … Σ 49, 392, 3136, 25088, 200704, 1605632, …
  (−∞, −1]nonnegativerational roots
• A132393 → basis A013611 → D
  4 · 12, 32 · 44, 192, 192 · 200, 1120, 1920, 1024 … Σ 0, 4, 44, 428, 4264, 45096, 511440, …
  (−∞, 0]nonnegative
• A038255 → basis A013609 → x ↦ x/(x+1)
  1 · 7, 9 · 49, 126, 81 · 343, 1323, 1701, 729 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  [−1, 0]nonnegativerational roots
• A201637 → reverse (shift +1) → θ+1
  1 · 0, 2 · 0, 4, 3 · 0, 12, 24, 4 … Σ 1, 2, 7, 40, 315, 3150, 38115, 540540, …
  (−∞, 0]nonnegative
• A038291 → basis A013611 → θ+1
  1 · 10, 8 · 100, 160, 48 · 1000, 2400, 1440, 256 … Σ 1, 18, 308, 5096, 82320, 1306144, …
  (−∞, 0]nonnegativerational roots
• A091042 → θ+1
  1 · 1, 6 · 1, 20, 15 · 1, 42, 105, 28 … Σ 1, 7, 36, 176, 832, 3840, 17408, 77824, …
  (−∞, 0]nonnegative
• A167884 → D
  1 · 18, 2 · 179, 358, 3 · 1636, 12172, 4908, 4 … Σ 0, 1, 20, 540, 18720, 795600, 40098240, …
  (−∞, 0]nonnegative
• A013611 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 25 · 1, 15, 75, 125 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  [−1, 0]nonnegativerational roots
• A038208 → D → D+1
  2 · 16, 8 · 72, 96, 24 · 256, 576, 384, 64 … Σ 0, 2, 24, 192, 1280, 7680, 43008, …
  (−∞, 0]nonnegativerational roots
• A060187 → x ↦ x+1
  1 · 2, 1 · 8, 8, 1 · 48, 72, 26, 1 … Σ 1, 3, 17, 147, 1697, 24483, 423857, …
  (−∞, −1]nonnegative
• A038220 → basis A013610 → θ+1
  1 · 5, 12 · 25, 120, 108 · 125, 900, 1620, 864 … Σ 1, 17, 253, 3509, 46585, 600281, …
  (−∞, 0]nonnegativerational roots
• A034870 → basis A013610 → D
  12, 18 · 96, 432, 648, 324 … Σ 0, 30, 1500, 56250, 1875000, 58593750, …
  (−∞, 0]nonnegativerational roots
• A225476 → θ+1
  1 · 1, 2 · 1, 8, 6 · 1, 26, 54, 24 … Σ 1, 3, 15, 105, 933, 10023, 126195, …
  (−∞, 0]nonnegative
• A026729 → basis A013611 → x ↦ x/(x+1)
  1 · 1, 5 · 2, 16, 30 · 3, 37, 145, 175 … Σ 1, 6, 48, 360, 2736, 20736, 157248, …
  [−1, 0]nonnegative
• A109692 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 8, 36, 40 · 48, 344, 816, 640 … Σ 1, 6, 84, 1848, 55440, 2106720, …
  [−1, 0]nonnegativerational roots
• A103633 → basis A013609 → x ↦ x/(x+1)
  1 · 1, 3 · 2, 10, 12 · 2, 16, 42, 36 … Σ 1, 4, 24, 96, 576, 2304, 13824, 55296, …
  [−1, 0]nonnegativerational roots
• A391533 → D
  4, 2 · 8, 10, 3 · 12, 26, 18, 4 · 16, 50, 57, 28, 5 … Σ 6, 21, 60, 156, 384, 912, 2112, 4800, …
  (−∞, 0]nonnegativerational roots
• A134082 → basis A007318 → x ↦ x/(x+1)
  1 · 3, 4 · 5, 16, 12 · 7, 36, 60, 32 … Σ 1, 7, 33, 135, 513, 1863, 6561, 22599, …
  [−1, 0]nonnegativerational roots
• A089231 → D+1
  1 · 3, 2 · 7, 18, 6 · 13, 84, 108, 24 … Σ 1, 5, 31, 229, 1961, 19081, 207775, …
  (−∞, 0]nonnegative
• A013611 → basis A013611 → D+1
  1 · 21, 16 · 185, 672, 256 · 1325, 8880, 16128, 4096 … Σ 1, 37, 1113, 30429, 787185, 19642581, …
  (−∞, 0]nonnegativerational roots
• A038338 → basis A013610 → x ↦ x+1
  3600, 4320, 1296 · 216000, 388800, 233280, 46656 … Σ 9216, 884736, 84934656, 8153726976, …
  (−∞, −1]nonnegativerational roots
• A145324 → basis A007318
  1 · 3, 2 · 12, 17, 6 · 60, 133, 98, 24 … Σ 1, 5, 35, 315, 3465, 45045, 675675, …
  (−∞, −1]nonnegativerational roots
• A038333 → x ↦ x+1 → D
  266, 98 · 7581, 5586, 1029 … Σ 364, 14196, 492128, 15994160, …
  (−∞, 0]nonnegativerational roots
• A225471 → basis A013609 → θ+1
  1 · 4, 4 · 32, 48, 12 · 384, 704, 288, 32 … Σ 1, 8, 92, 1408, 27024, 624768, …
  (−∞, 0]nonnegative
• A038305 → x ↦ x+1 → D
  78, 18 · 1521, 702, 81 · 26364, 18252, 4212, 324 … Σ 96, 2304, 49152, 983040, 18874368, …
  (−∞, 0]nonnegativerational roots
• A144438 → D+1
  1 · 2, 1 · 6, 7, 1 · 15, 42, 17, 1 … Σ 1, 3, 14, 75, 471, 3402, 27844, 254970, …
  (−∞, 0]nonnegative
• A249253 → θ+1
  1 · 3, 2 · 12, 10 · 60, 56, 3 … Σ 1, 5, 22, 119, 753, 5479, 45136, …
  (−∞, 0]nonnegative
• A136572 → θ+1
  1 · 0, 2 · 0, 0, 6 · 0, 0, 0, 24 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  [−1, 0]nonnegativerational roots
• A086915 → θ+1
  2 · 4, 8 · 12, 48, 24 · 48, 288, 288, 64 … Σ 2, 12, 84, 688, 6480, 69024, 820064, …
  (−∞, 0]nonnegative
• A038279 → basis A013611
  1 · 9, 4 · 81, 72, 16 · 729, 972, 432, 64 … Σ 1, 13, 169, 2197, 28561, 371293, …
  (−∞, −1]nonnegativerational roots
• A372873 → x ↦ x+1
  1 · 2, 2 · 5, 9, 4 · 14, 36, 30, 8 … Σ 1, 4, 18, 88, 452, 2384, 12744, 68576, …
  (−∞, −1]nonnegative
• A038333 → basis A013609 → x ↦ x+1
  1089, 924, 196 · 35937, 45738, 19404, 2744 … Σ 2209, 103823, 4879681, 229345007, …
  (−∞, −1]nonnegativerational roots
• A090582 → D
  1 · 6, 2 · 36, 28, 3 · 240, 300, 90, 4 … Σ 0, 1, 8, 67, 634, 6793, 81780, 1095967, …
  (−∞, 0]nonnegative