Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038328 → basis A013609 → θ+1
  196, 224, 48 · 2744, 4704, 2016, 256 … Σ 468, 9720, 198288, 3989088, 79361856, …
  (−∞, 0]nonnegativerational roots
• A027465 → basis A013611
  1 · 4, 4 · 16, 32, 16 · 64, 192, 192, 64 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, 0]nonnegativerational roots
• A131431 → x ↦ x+1 → basis A007318
  1 · 8, 4 · 28, 28, 7 · 80, 120, 60, 10 … Σ 1, 12, 63, 270, 1053, 3888, 13851, …
  (−∞, −1]nonnegativerational roots
• Veronese parent of A098172 → Veronese 3 (j=1) → x ↦ x/(x+1)
  1 · 0, 1 · 0, 0, 1 · 0, 0, 1, 2 … Σ 1, 1, 1, 3, 9, 21, 45, 99, …
  [−1, 0]nonnegative
• A008306 → (θ+1)²
  1 · 2 · 6, 12 · 24, 80 … Σ 1, 2, 18, 104, 775, 6306, 57372, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013610 → (θ+1)²
  1 · 1, 24, 81 · 1, 48, 486, 1728, 2025 … Σ 1, 106, 4288, 128512, 3309568, …
  (−∞, 0]nonnegative
• A033282 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 5 · 11, 37, 31 · 45, 228, 384, 215 … Σ 1, 8, 79, 872, 10306, 127568, 1632619, …
  [−1, −1/2)nonnegative
• A038281 → basis A013611
  121, 264, 144 · 1331, 4356, 4752, 1728 … Σ 529, 12167, 279841, 6436343, 148035889, …
  (−∞, 0]nonnegativerational roots
• A038259 → basis A013610
  1 · 11, 15 · 121, 330, 225 · 1331, 5445, 7425, 3375 … Σ 1, 26, 676, 17576, 456976, 11881376, …
  [−1, 0]nonnegativerational roots
• A154233 → D
  1 · 7, 2 · 19, 38, 3 · 39, 342, 117, 4 … Σ 0, 1, 9, 60, 502, 4155, 40845, 415240, …
  (−∞, 0]nonnegative
• A036969 → (θ+1)²
  1 · 1, 4 · 1, 20, 9 · 1, 84, 126, 16 … Σ 1, 5, 30, 227, 2169, 25480, 358993, …
  (−∞, 0]nonnegative
• A089503 → x ↦ x+1
  1 · 5, 4 · 43, 72, 30 · 529, 1368, 1176, 336 … Σ 1, 9, 145, 3409, 105921, 4102681, …
  (−∞, −1]nonnegative
• A207606 → basis A013609 → D
  22, 16 · 88, 136, 48 · 310, 744, 552, 128 … Σ 38, 272, 1734, 10372, 59584, 332864, …
  (−∞, 0]nonnegative
• A127648 → D → θ+1
  2 · 0, 12 · 0, 0, 36 · 0, 0, 0, 80 … Σ 0, 0, 2, 12, 36, 80, 150, 252, …
  (−∞, 0]nonnegativerational roots
• A038279 → basis A013609 → θ+1
  1 · 9, 4 · 81, 72, 12 · 729, 972, 324, 32 … Σ 1, 13, 165, 2057, 25289, 307461, …
  (−∞, 0]nonnegativerational roots
• A013612 → basis A013609 → θ+1
  36, 240, 300 · 216, 2160, 5400, 4000 … Σ 576, 11776, 229376, 4325376, 79691776, …
  (−∞, 0]nonnegativerational roots
• A391533 → Pₙ + Pₙ₊₁ → reverse
  1, 6, 12, 8 · 1, 7, 18, 20, 8 · 1, 8, 25, 38, 28, 8 … Σ 27, 54, 108, 216, 432, 864, 1728, 3456, …
  (−∞, 0]nonnegativerational roots
• A085880 → basis A013611 → x ↦ x+1
  1 · 6, 4 · 72, 96, 32 · 1080, 2160, 1440, 320 … Σ 1, 10, 200, 5000, 140000, 4200000, …
  (−∞, −1]nonnegativerational roots
• A038327 → basis A013609 → reverse
  1 · 2, 13 · 4, 52, 169 · 8, 156, 1014, 2197 … Σ 1, 15, 225, 3375, 50625, 759375, …
  (−∞, 0]nonnegativerational roots
• A038303 → basis A013609 → reverse
  1 · 2, 11 · 4, 44, 121 · 8, 132, 726, 1331 … Σ 1, 13, 169, 2197, 28561, 371293, …
  (−∞, 0]nonnegativerational roots
• A027465 → D
  1 · 6, 2 · 27, 18, 3 · 108, 108, 36, 4 … Σ 0, 1, 8, 48, 256, 1280, 6144, 28672, …
  (−∞, −1]nonnegativerational roots
• A027467 → basis A013611 → θ+1
  1 · 16, 8 · 256, 256, 48 · 4096, 6144, 2304, 256 … Σ 1, 24, 560, 12800, 288000, 6400000, …
  (−∞, 0]nonnegativerational roots
• image under θ+1 → (θ+1)² → x ↦ x+1
  1 · 8, 8 · 27, 54, 27 · 64, 192, 192, 64 … Σ 1, 16, 108, 512, 2000, 6912, 21952, …
  (−∞, −1]nonnegativerational roots
• A154929 → θ+1
  1 · 2, 2 · 3, 8, 3 · 5, 20, 18, 4 … Σ 1, 4, 14, 47, 152, 480, 1488, 4548, …
  (−∞, 0]nonnegative
• A124932 → reverse → x ↦ x+1
  1 · 5, 2 · 18, 15, 3 · 56, 72, 30, 4 … Σ 1, 7, 36, 162, 675, 2673, 10206, 37908, …
  (−∞, −1]nonnegativerational roots
• A113953 → x ↦ x−10
  1 · -10, 1 · 80, -18, 1 · -600, 220, -26, 1 … Σ 1, -9, 63, -405, 2511, -15309, 92583, …
  (2, 10]
• A081577 → D
  1 · 4, 2 · 7, 14, 3 · 10, 44, 30, 4 … Σ 0, 1, 6, 24, 88, 300, 984, 3136, …
  (−∞, 0]nonnegative
• A166340 → D+1
  9, 10, 1 · 20, 57, 22, 1 · 43, 270, 240, 46, 1 … Σ 20, 100, 600, 4200, 33600, 302400, …
  (−∞, 0]nonnegative
• A062190 → reverse → θ+1
  1 · 6, 2 · 21, 28, 3 · 56, 168, 72, 4 … Σ 1, 8, 52, 300, 1595, 8008, 38584, …
  (−∞, 0]nonnegative
• A178822 → x ↦ x+1 → D+1
  1 · 18, 6 · 168, 126, 21 · 1120, 1344, 504, 56 … Σ 1, 24, 315, 3024, 23814, 163296, …
  (−∞, 0]nonnegativerational roots
• A286718 → basis A013611 → D
  4 · 28, 32 · 264, 480, 192 · 3224, 7392, 4992, 1024 … Σ 0, 4, 60, 936, 16632, 338256, 7803216, …
  (−∞, 0]nonnegative
• A038291 → basis A013611 → (θ+1)²
  1 · 10, 16 · 100, 320, 144 · 1000, 4800, 4320, 1024 … Σ 1, 26, 564, 11144, 207760, 3720864, …
  (−∞, 0]nonnegative
• A145324 → D
  2 · 5, 12 · 9, 52, 72 · 14, 142, 462, 480 … Σ 0, 2, 17, 133, 1098, 9846, 96408, …
  (−∞, 0]nonnegative
• A105278 → D
  1 · 6, 2 · 36, 24, 3 · 240, 240, 60, 4 … Σ 0, 1, 8, 63, 544, 5225, 55656, 653023, …
  (−∞, 0]nonnegative
• A062196 → reverse → D
  1 · 8, 2 · 30, 30, 3 · 80, 180, 72, 4 … Σ 0, 1, 10, 63, 336, 1650, 7722, 35035, …
  (−∞, 0]nonnegative
• A120910 → Veronese 2 (j=1) → D
  3 · 24 · 120, 6 · 480, 72 … Σ 0, 0, 0, 3, 24, 126, 552, 2193, …
  (−∞, 0]nonnegative
• A204420 → D+1
  1 · 1, 1 · 6, 12, 3 · 120, 300, 135, 15 … Σ 1, 2, 21, 570, 29505, 2489130, …
  (−∞, 0]nonnegative
• A056241 → basis A007318
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … Σ 1, 3, 11, 45, 193, 843, 3707, 16341, …
  (−∞, −1]nonnegative
• A356546 → x ↦ x+1 → θ+1
  1 · 4, 4 · 24, 48, 18 · 160, 480, 360, 80 … Σ 1, 8, 90, 1080, 13230, 163296, 2020788, …
  (−∞, 0]nonnegativerational roots
• A130534 → D → θ+1
  1 · 3, 4 · 11, 24, 9 · 50, 140, 90, 16 … Σ 0, 1, 7, 44, 296, 2204, 18236, 166984, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → D → θ+1
  -1 · 2, 4 · -3, -12, -9 · 4, 24, 36, 16 … Σ 0, -1, 6, -24, 80, -240, 672, -1792, …
  (−∞, 0]rational roots
• A056242 → θ+1 → reverse
  1 · 4, 1 · 12, 10, 1 · 32, 48, 18, 1 … Σ 1, 5, 23, 99, 408, 1632, 6388, 24596, …
  (−∞, 0]nonnegative
• A143495 → reverse → x ↦ x+1
  1 · 4, 3 · 17, 25, 9 · 77, 167, 118, 27 … Σ 1, 7, 51, 389, 3121, 26423, 236403, …
  (−∞, −1]nonnegative
• A081580 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 9, 9 · 1, 16, 42, 28 … Σ 1, 3, 19, 87, 451, 2223, 11179, 55767, …
  [−1, 0]nonnegative
• A122848: Exponential Riordan array (1, x(1+x/2)).
  1 · 0, 1 · 0, 1, 1 · 0, 0, 3, 1 … Σ 1, 1, 2, 4, 10, 26, 76, 232, …
  (−∞, 0]nonnegative
• A108838 → D → θ+1
  2 · 8, 8 · 20, 60, 18 · 40, 240, 216, 32 … Σ 0, 2, 16, 98, 528, 2640, 12584, 58058, …
  (−∞, 0]nonnegative
• A144438 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 17, 45, 30 … Σ 1, 3, 15, 93, 703, 6285, 64983, 763533, …
  [−1, 0]nonnegative
• A290319 → x ↦ x+1 → D+1
  1 · 3, 1 · 20, 10, 1 · 212, 128, 21, 1 … Σ 1, 4, 31, 362, 5661, 111024, 2619387, …
  (−∞, 0]nonnegative
• A109692 → basis A007318 → basis A007318
  1 · 3, 1 · 21, 16, 3 · 231, 281, 113, 15 … Σ 1, 4, 40, 640, 14080, 394240, 13404160, …
  (−∞, −1]nonnegativerational roots
• A028421 → θ+1
  1 · 1, 4 · 2, 12, 9 · 6, 44, 54, 16 … Σ 1, 5, 23, 120, 724, 5012, 39332, …
  (−∞, 0]nonnegative