Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A290596 → D+1
  1 · 3, 1 · 20, 12, 1 · 200, 168, 27, 1 … Σ 1, 4, 33, 396, 6165, 117396, 2636469, …
  (−∞, 0]nonnegative
• A159854 → Veronese 2 → (θ+1)²
  0, 4 · 0, 4 · 0, 4, 9 · 0, 4, 27 … Σ 4, 4, 13, 31, 74, 174, 404, 928, …
  (−∞, 0]nonnegative
• A113413 → basis A013609 → reverse
  1 · 2, 3 · 4, 12, 7 · 8, 36, 46, 17 … Σ 1, 5, 23, 107, 497, 2309, 10727, 49835, …
  (−∞, 0]nonnegative
• A136124 → D+1
  1 · 3, 1 · 11, 7, 1 · 50, 44, 12, 1 … Σ 1, 4, 19, 107, 702, 5274, 44712, …
  (−∞, −1]nonnegative
• A027467 → basis A013610 → θ+1
  1 · 16, 6 · 256, 192, 27 · 4096, 4608, 1296, 108 … Σ 1, 22, 475, 10108, 212629, 4430914, …
  (−∞, 0]nonnegativerational roots
• A038271 → basis A013610 → basis A007318
  1 · 27, 15 · 729, 810, 225 … Σ 1, 42, 1764, 74088, 3111696, 130691232, …
  (−∞, −1]nonnegativerational roots
• A130123 → Veronese 2 → D+1
  1 · 4, 4 · 0, 32, 16 · 0, 0, 192, 64 … Σ 1, 0, 8, 0, 48, 0, 256, 0, …
  (−∞, 0]nonnegativerational roots
• A099093 → basis A013609 → D+1
  1 · 9, 6 · 54, 114, 36 · 279, 1026, 1044, 216 … Σ 1, 15, 204, 2565, 30807, 358668, …
  (−∞, 0]nonnegative
• A199478 → x ↦ x+1
  2, 1 · 4, 3 · 8, 9, 2 · 16, 23, 8 … Σ 3, 7, 19, 47, 123, 311, 803, 2047, …
  (−∞, −1]nonnegative
• A357613 → reverse
  1 · 2, 3 · 6, 20, 15 · 20, 105, 168, 84 … Σ 1, 5, 41, 377, 3649, 36365, 369305, …
  (−∞, 0]nonnegative
• A274498 → D+1
  1 · 3 · 9, 6 · 27, 18 … Σ 1, 3, 15, 45, 189, 567, 2187, 6561, …
  (−∞, 0]nonnegativerational roots
• A113953 → basis A013610 → D
  3 · 12, 18 · 33, 126, 81 · 90, 504, 810, 324 … Σ 0, 3, 30, 240, 1728, 11712, 76416, …
  (−∞, 0]nonnegative
• A038335 → x ↦ x+1
  1 · 21, 9 · 441, 378, 81 · 9261, 11907, 5103, 729 … Σ 1, 30, 900, 27000, 810000, 24300000, …
  (−∞, −1]nonnegativerational roots
• A035607 → D
  2 · 4, 4 · 6, 16, 6 · 8, 36, 36, 8 … Σ 0, 0, 2, 8, 28, 88, 262, 752, …
  (−∞, 0]nonnegative
• one_plus_derivative of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
  1 · 3, 4, 1 · 7, 26, 28, 10, 1 … Σ 1, 8, 72, 768, 9600, 138240, 2257920, …
  (−∞, 0]nonnegative
• basis-map coefficients of A371080, shifted → basis A008277
  1 · 1, 1 · 4, 6, 1 · 28, 52, 15, 1 … Σ 1, 2, 11, 96, 1141, 17126, 310207, …
  (−∞, 0]nonnegative
• A048993 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 4, 27, 16 … Σ 1, 4, 13, 47, 188, 825, 3937, 20270, …
  (−∞, 0]nonnegative
• A367270 → D+1
  1 · 1 · 3, 2 · 7, 12, 3 … Σ 1, 1, 5, 22, 95, 406, 1722, 7260, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
  1, 4, 3 · 1, 6, 9, 4 · 1, 8, 18, 16, 5 … Σ 8, 20, 48, 112, 256, 576, 1280, 2816, …
  (−∞, 0]nonnegativerational roots
• A178820 → D+1
  1 · 8, 4 · 30, 40, 10 · 80, 180, 120, 20 … Σ 1, 12, 80, 400, 1680, 6272, 21504, …
  (−∞, 0]nonnegativerational roots
• A144441 → x ↦ x+1
  1 · 2, 1 · 16, 16, 1 · 168, 252, 86, 1 … Σ 1, 3, 33, 507, 10537, 278067, 8917809, …
  (−∞, −1]nonnegative
• basis-map coefficients of A371080, shifted → θ+1
  1 · 0, 2 · 0, 6, 3 · 0, 36, 27, 4 … Σ 1, 2, 9, 67, 698, 9354, 153304, …
  (−∞, 0]nonnegative
• A102547 → basis A007318
  1 · 1 · 1 · 2, 1 … Σ 1, 1, 1, 3, 5, 7, 13, 23, …
  (−∞, −1]nonnegative
• basis-map coefficients of A371080, shifted → reverse
  1 · 1 · 1, 3 · 1, 9, 18 … Σ 1, 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0]nonnegativerational roots
• A059364 → basis A007318
  1 · 3, 1 · 15, 11, 2 · 105, 122, 47, 6 … Σ 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, −1]nonnegativerational roots
• A137597 → reverse → θ+1
  1 · 1, 4 · 1, 10, 15 · 1, 18, 66, 60 … Σ 1, 5, 26, 145, 868, 5559, 37930, …
  (−∞, 0]nonnegative
• A124324 → D+1
  1 · 1 · 2, 1 · 5, 4 … Σ 1, 1, 3, 9, 32, 128, 565, 2719, …
  (−∞, 0]nonnegative
• A013609 → basis A013610 → D
  6 · 36, 72 · 162, 648, 648 · 648, 3888, 7776, 5184 … Σ 0, 6, 108, 1458, 17496, 196830, …
  (−∞, 0]nonnegativerational roots
• A038255 → basis A013611 → reverse
  1 · 4, 7 · 16, 56, 49 · 64, 336, 588, 343 … Σ 1, 11, 121, 1331, 14641, 161051, …
  (−∞, 0]nonnegativerational roots
• A039756 → reverse → D+1
  1 · 2, 1 · 5, 6, 1 · 14, 31, 12, 1 … Σ 1, 3, 12, 58, 324, 2044, 14320, 109960, …
  (−∞, 0]nonnegative
• A367270 → basis A013611 → x ↦ x+1
  1 · 1 · 11, 8 · 106, 144, 48 … Σ 1, 1, 19, 298, 4483, 67006, 1004590, …
  (−∞, −1]nonnegative
• A030528 → x ↦ x−15
  1 · -14, 1 · 195, -28, 1 · -2715, 586, -42, 1 … Σ 1, -13, 168, -2170, 28028, -362012, …
  (11, 15]
• A395972 → x ↦ x+1
  28, 32, 8 · 272, 504, 288, 48 … Σ 68, 1112, 23568, 610912, 18674752, …
  (−∞, −1]nonnegative
• A038334 → (θ+1)²
  144, 768, 576 · 1728, 13824, 20736, 8192 … Σ 1488, 44480, 1235200, 32640000, …
  (−∞, 0]nonnegative
• A143491 → basis A013609 → (θ+1)²
  12, 56, 36 · 60, 376, 432, 128 … Σ 104, 996, 10084, 109364, 1274552, …
  (−∞, 0]nonnegative
• A274496 → D+1
  1 · 2 · 4, 2 · 8, 4 … Σ 1, 2, 6, 12, 32, 64, 160, 320, …
  (−∞, 0]nonnegativerational roots
• A112486 → x ↦ x+1
  1 · 2, 1 · 10, 11, 3 · 82, 141, 80, 15 … Σ 1, 3, 24, 318, 5892, 140304, 4082712, …
  [−2, −1)nonnegative
• A257609 → D (shift +2) → reverse
  3, 22, 11 · 4, 78, 132, 26 · 5, 228, 906, 604, 57 … Σ 36, 240, 1800, 15120, 141120, 1451520, …
  (−∞, 0]nonnegative
• A173018 → D+1
  1 · 1 · 2, 1 · 5, 6, 1 … Σ 1, 1, 3, 12, 60, 360, 2520, 20160, …
  (−∞, 0]nonnegative
• A038277 → basis A013609 → D+1
  1 · 40, 22 · 1116, 1760, 484 … Σ 1, 62, 3360, 169600, 8192000, …
  (−∞, 0]nonnegativerational roots
• A196347 → basis A013611 → D+1
  1 · 6, 4 · 40, 96, 32 · 336, 1440, 1728, 384 … Σ 1, 10, 168, 3888, 114048, 4043520, …
  (−∞, 0]nonnegativerational roots
• A038327 → x ↦ x/(x+1)
  1 · 12, 13 · 144, 312, 169 · 1728, 5616, 6084, 2197 … Σ 1, 25, 625, 15625, 390625, 9765625, …
  [−1, 0]nonnegativerational roots
• A201730 → basis A007318
  1 · 2 · 6, 1 · 20, 6 … Σ 1, 2, 7, 26, 97, 362, 1351, 5042, …
  (−∞, −1]nonnegative
• A075502 → D+1
  70, 23, 1 · 686, 427, 45, 1 · 7546, 7595, 1435, 74, 1 … Σ 94, 1159, 16651, 271550, 4918823, …
  (−∞, 0]nonnegative
• A081577 → basis A013611 → D
  4 · 24, 32 · 96, 320, 192 · 352, 1856, 2688, 1024 … Σ 0, 4, 56, 608, 5920, 54192, 477024, …
  (−∞, 0]nonnegative
• A038231 → Veronese 2 → θ+1
  1 · 4 · 16, 2 · 64, 24 … Σ 1, 4, 18, 88, 451, 2364, 12500, 66224, …
  (−∞, 0]nonnegative
• A249250 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 4, 7 · 10, 32, 25 · 28, 132, 204, 103 … Σ 1, 11, 67, 467, 3139, 21299, 144163, …
  [−1, 0]nonnegative
• A038333 → basis A013611 → x ↦ x+1
  2209, 2632, 784 · 103823, 185556, 110544, 21952 … Σ 5625, 421875, 31640625, 2373046875, …
  (−∞, −1]nonnegativerational roots
• A136124 → basis A013611 → θ+1
  1 · 3, 8 · 12, 56, 48 · 60, 376, 576, 256 … Σ 1, 11, 116, 1268, 14696, 181816, …
  (−∞, 0]nonnegative
• A249252 → D
  4 · 8, 16 · 12, 64, 48 · 16, 144, 288, 128 … Σ 0, 4, 24, 124, 576, 2532, 10728, 44300, …
  (−∞, 0]nonnegative