Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A131431 → (θ+1)² → θ+1
  1 · 0, 32 · 0, 0, 189 · 0, 0, 0, 640 … Σ 1, 32, 189, 640, 1625, 3456, 6517, …
  (−∞, 0]nonnegativerational roots
• A292219 → D+1
  1 · 7, 1 · 80, 22, 1 · 1260, 504, 45, 1 … Σ 1, 8, 103, 1810, 40109, 1069756, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A008288 → D
  3, 2 · 7, 26, 21, 4 · 11, 82, 189, 164, 55, 6 … Σ 0, 5, 58, 507, 3940, 28705, 200766, …
  (−∞, 0]nonnegative
• A038301 → reverse → x ↦ x/(x+1)
  121, 440, 400 · 1331, 7260, 13200, 8000 … Σ 961, 29791, 923521, 28629151, …
  [−1, 0]nonnegativerational roots
• A397048 → D
  1 · 5, 2 · 37, 24, 3 · 369, 302, 66, 4 … Σ 0, 1, 7, 64, 741, 10451, 174266, …
  (−∞, 0]nonnegative
• A008955 → basis A013609 → D
  2 · 26, 32 · 440, 1256, 864 … Σ 0, 2, 58, 2560, 172032, 16641632, …
  (−∞, 0]nonnegative
• delannoy → basis A013609 → basis A007318
  1 · 4, 2 · 19, 18, 4 · 88, 124, 56, 8 … Σ 1, 6, 41, 276, 1861, 12546, 84581, …
  (−∞, −1]nonnegative
• A126351 → x ↦ x+1 → D+1
  1 · 5, 2 · 23, 21, 4 · 108, 157, 67, 8 … Σ 1, 7, 48, 340, 2551, 20473, 175954, …
  (−∞, 0]nonnegative
• A166979 → x ↦ x/(x+1)
  1, 13, 13 · 1, 114, 255, 143 … Σ 27, 513, 11745, 299619, 8656875, …
  [−1, 0]nonnegative
• A111006 → D
  1 · 1, 4 · 0, 4, 9 · 0, 2, 15, 20 … Σ 0, 1, 5, 13, 37, 89, 217, 501, …
  (−∞, 0]nonnegative
• A089627 → D
  2 · 6 · 12, 12 · 20, 60 … Σ 0, 0, 2, 6, 24, 80, 270, 882, …
  (−∞, 0]nonnegative
• A038333 → basis A013610 → reverse
  441, 798, 361 · 9261, 25137, 22743, 6859 … Σ 1600, 64000, 2560000, 102400000, …
  (−∞, 0]nonnegativerational roots
• A111125 → basis A013610 → reverse
  1 · 3, 4 · 9, 21, 11 · 27, 90, 93, 29 … Σ 1, 7, 41, 239, 1393, 8119, 47321, …
  (−∞, 0]nonnegative
• A046739 → x ↦ x+1
  9, 9, 1 · 44, 66, 24, 1 · 265, 530, 320, 55, 1 … Σ 19, 135, 1171, 11823, 136459, 1771815, …
  (−∞, −1]nonnegative
• A137320 → x ↦ x+1 → x ↦ x+1
  1 · 4, 2 · 20, 18, 4 · 120, 148, 60, 8 … Σ 1, 6, 42, 336, 3024, 30240, 332640, …
  (−∞, −1]nonnegativerational roots
• A056242 → reverse → x ↦ x+1
  1 · 3, 1 · 10, 7, 1 · 34, 37, 12, 1 … Σ 1, 4, 18, 84, 396, 1872, 8856, 41904, …
  (−∞, −1]nonnegative
• A008306 → basis A007318
  1 · 2 · 9, 3 · 44, 20 … Σ 1, 2, 12, 64, 440, 3408, 30016, 294656, …
  (−∞, −1]nonnegative
• A082985 → reverse → θ+1
  1 · 3, 2 · 5, 10, 3 · 7, 28, 21, 4 … Σ 1, 5, 18, 60, 191, 589, 1775, 5257, …
  (−∞, 0]nonnegative
• A116412 → (θ+1)²
  1 · 3, 4 · 6, 24, 9 · 12, 84, 81, 16 … Σ 1, 7, 39, 193, 886, 3870, 16303, 66811, …
  (−∞, 0]nonnegative
• A257610 → (θ+1)²
  1 · 2, 8 · 4, 80, 36 · 8, 528, 1188, 128 … Σ 1, 10, 120, 1852, 34384, 744240, …
  (−∞, 0]nonnegative
• A145597 → D
  3 · 15, 12 · 45, 90, 30 · 105, 378, 315, 60 … Σ 0, 3, 27, 165, 858, 4095, 18564, 81396, …
  (−∞, 0]nonnegative
• A128813 → θ+1
  1 · 1, 2 · 1, 8, 9 · 1, 20, 81, 72 … Σ 1, 3, 18, 174, 2474, 48824, 1281104, …
  (−∞, 0]nonnegative
• A159854 → D
  0, 2 · 0, 2, 3 · 0, 2, 6, 4 · 0, 2, 9, 12, 5 … Σ 2, 5, 12, 28, 64, 144, 320, 704, …
  (−∞, 0]nonnegativerational roots
• A089627 → D+1
  1 · 1 · 3, 2 · 7, 6 … Σ 1, 1, 5, 13, 43, 131, 411, 1275, …
  (−∞, 0]nonnegative
• A038333 → basis A013609 → D
  532, 392 · 15162, 22344, 8232 … Σ 924, 45738, 2012472, 83014470, …
  (−∞, 0]nonnegativerational roots
• A116414 → basis A013610 → D
  3 · 30, 18 · 207, 270, 81 · 1230, 2538, 1620, 324 … Σ 0, 3, 48, 558, 5712, 54609, 500256, …
  (−∞, 0]nonnegative
• A143685 → γ → D+1
  1 · 1 · 10, 9 · 19, 18 … Σ 1, 1, 19, 37, 298, 802, 4465, 14203, …
  (−∞, 0]nonnegative
• A038259 → basis A013610 → x ↦ x+1
  1 · 26, 15 · 676, 780, 225 … Σ 1, 41, 1681, 68921, 2825761, 115856201, …
  (−∞, −1]nonnegativerational roots
• A038327 → x ↦ x+1 → θ+1
  1 · 13, 2 · 169, 52, 3 · 2197, 1014, 117, 4 … Σ 1, 15, 224, 3332, 49392, 729904, …
  (−∞, 0]nonnegativerational roots
• A113953 → θ+1
  1 · 0, 2 · 0, 4, 3 · 0, 0, 12, 4 … Σ 1, 2, 7, 16, 41, 94, 219, 492, …
  (−∞, 0]nonnegative
• A178821 → D
  5 · 30, 30 · 105, 210, 105 · 280, 840, 840, 280 … Σ 0, 5, 60, 420, 2240, 10080, 40320, …
  (−∞, 0]nonnegativerational roots
• A111125 → D → (θ+1)²
  1 · 5, 8 · 14, 56, 27 · 30, 216, 243, 64 … Σ 0, 1, 13, 97, 553, 2688, 11760, 47747, …
  (−∞, 0]nonnegative
• A054142 → Veronese 2 → x ↦ x+1
  1 · 1 · 2, 1 · 7, 6 … Σ 1, 1, 3, 13, 35, 117, 379, 1197, …
  (−∞, −1]nonnegative
• A038259 → (θ+1)²
  1 · 6, 20 · 36, 240, 225 · 216, 2160, 4050, 2000 … Σ 1, 26, 501, 8426, 130801, 1924626, …
  (−∞, 0]nonnegative
• A046716 → basis A013609 → θ+1
  1 · 2, 4 · 5, 20, 12 · 16, 100, 132, 32 … Σ 1, 6, 37, 280, 2901, 40882, 730009, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → x ↦ x+1
  1 · -2, -1 · 4, 4, 1 · -8, -12, -6, -1 … Σ 1, -3, 9, -27, 81, -243, 729, -2187, …
  (−∞, −1]rational roots
• A038277 → D
  11 · 154, 242 · 1617, 5082, 3993 … Σ 0, 11, 396, 10692, 256608, 5773680, …
  (−∞, 0]nonnegativerational roots
• A139548 → basis A013611
  1 · 8, 32, 32 · 64, 512, 1536, 2048, 1024 … Σ 1, 72, 5184, 373248, 26873856, …
  [−1, 0]nonnegativerational roots
• A034870 → basis A008277
  1 · 4, 5, 1 · 16, 65, 55, 14, 1 … Σ 1, 10, 151, 3263, 94828, 3535027, …
  (−∞, 0]nonnegative
• A159854 → D+1
  0, 2, 1 · 0, 2, 4, 1 · 0, 2, 7, 6, 1 … Σ 3, 7, 16, 36, 80, 176, 384, 832, …
  (−∞, 0]nonnegative
• A130534 → D²∘x (shift +1) → x ↦ x/(x+1)
  1 · 3, 6 · 11, 40, 35 · 50, 255, 420, 225 … Σ 1, 9, 86, 950, 12139, 177331, 2924172, …
  [−1, 0]nonnegative
• A038323 → x ↦ x+1 → reverse
  1 · 9, 20 · 81, 360, 400 · 729, 4860, 10800, 8000 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  (−∞, 0]nonnegativerational roots
• A128099 → D
  2 · 4 · 6, 8 · 8, 24 … Σ 0, 0, 2, 4, 14, 32, 82, 188, …
  (−∞, 0]nonnegative
• A102365 → D+1
  1 · 1 · 2, 1 · 6, 7, 1 … Σ 1, 1, 3, 14, 86, 657, 5995, 63541, …
  (−∞, 0]nonnegative
• A059364 → basis A007318 → D+1
  1 · 4, 1 · 26, 15, 2 · 227, 216, 65, 6 … Σ 1, 5, 43, 514, 7802, 143032, 3067048, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → Veronese 2 → D+1
  1 · 2, 2 · 0, 48, 24 · 0, 0, 2160, 720 … Σ 1, 0, 4, 0, 72, 0, 2880, 0, …
  (−∞, 0]nonnegativerational roots
• A133087 → Veronese 2 → θ+1
  1 · 2 · 1, 2 · 2, 8 … Σ 1, 2, 3, 10, 16, 52, 80, 256, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → basis A013610 → reverse
  1 · -3, -1 · 9, 6, 1 · -27, -27, -9, -1 … Σ 1, -4, 16, -64, 256, -1024, 4096, …
  (−∞, 0]rational roots
• A099097 → x ↦ x−6
  1 · -18, 3 · 318, -107, 9 · -5616, 2844, -480, 27 … Σ 1, -15, 220, -3225, 47275, -693000, …
  (50/9, 6]
• A142458 → θ+1
  1 · 1, 2 · 1, 16, 3 · 1, 78, 117, 4 … Σ 1, 3, 20, 200, 2640, 43120, 837760, …
  (−∞, 0]nonnegative