Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A166962 → D+1
  1 · 2, 1 · 13, 14, 1 · 104, 241, 72, 1 … Σ 1, 3, 28, 418, 9448, 293072, 11811692, …
  (−∞, 0]nonnegative
• A086645 → basis A013610 → (θ+1)²
  1 · 2, 12 · 8, 96, 81 · 32, 576, 1458, 432 … Σ 1, 14, 185, 2498, 32009, 389390, …
  (−∞, 0]nonnegative
• A013622 → x ↦ x/(x+1)
  1 · 3, 8 · 9, 48, 64 · 27, 216, 576, 512 … Σ 1, 11, 121, 1331, 14641, 161051, …
  [−1, 0]nonnegativerational roots
• A200536 → basis A013609 → θ+1
  1 · 6, 28, 24 · 36, 336, 876, 896, 320 … Σ 1, 58, 2464, 92512, 3248896, 109408768, …
  (−∞, 0]nonnegative
• A257624 → D
  5 · 80, 50 · 915, 1830, 375 … Σ 0, 5, 130, 3120, 79040, 2173600, …
  (−∞, 0]nonnegative
• A075505 → reverse
  1, 30, 100 · 1, 60, 700, 1000 … Σ 131, 1761, 27601, 506651, 10674211, …
  (−∞, 0]nonnegative
• A085880 → basis A013611 → θ+1
  1 · 2, 8 · 8, 64, 96 · 40, 480, 1440, 1280 … Σ 1, 10, 168, 3240, 66528, 1415232, …
  (−∞, 0]nonnegativerational roots
• A103327 → basis A013611 → (θ+1)²
  1 · 4, 16 · 16, 192, 144 · 64, 1280, 3456, 1024 … Σ 1, 20, 352, 5824, 89344, 1290240, …
  (−∞, 0]nonnegative
• A161198 → basis A013609 → basis A007318
  1 · 7, 4 · 63, 64, 16 · 693, 956, 432, 64 … Σ 1, 11, 143, 2145, 36465, 692835, …
  (−∞, −1]nonnegativerational roots
• A154986 → γ → D+1
  1 · 1 · 3, 2 · 9, 8 … Σ 1, 1, 5, 17, 113, 633, 5373, 41409, …
  (−∞, 0]nonnegative
• A137320 → basis A013609 → basis A007318
  1 · 6, 4 · 42, 52, 16 · 336, 584, 336, 64 … Σ 1, 10, 110, 1320, 17160, 240240, …
  (−∞, −1]nonnegativerational roots
• A038301 → D+1
  279, 440, 121 · 3402, 9207, 7260, 1331 … Σ 840, 21200, 512000, 12000000, …
  (−∞, 0]nonnegativerational roots
• A367270 → (θ+1)²
  1 · 1 · 1, 8 · 1, 24, 27 … Σ 1, 1, 9, 52, 275, 1386, 6762, 32208, …
  (−∞, 0]nonnegative
• A062196 → θ+1
  1 · 1, 6 · 1, 16, 18 · 1, 30, 90, 40 … Σ 1, 7, 35, 161, 714, 3102, 13299, 56485, …
  (−∞, 0]nonnegative
• A013620 → Veronese 2 → x ↦ x+1
  1 · 2 · 13, 9 · 62, 54 … Σ 1, 2, 22, 116, 772, 4712, 29656, …
  (−∞, −1]nonnegative
• A172431 → reverse → D+1
  1 · 3, 1 · 7, 6, 1 · 14, 22, 9, 1 … Σ 1, 4, 14, 46, 145, 444, 1331, 3926, …
  (−∞, 0]nonnegative
• A049029 → x ↦ x/(x+1)
  1 · 5, 6 · 45, 105, 61 · 585, 2010, 2295, 871 … Σ 1, 11, 211, 5761, 203921, 8865211, …
  [−1, 0]nonnegative
• A013611 → D → θ+1
  4 · 8, 64 · 12, 192, 576 · 16, 384, 2304, 4096 … Σ 0, 4, 72, 780, 6800, 52500, 375000, …
  (−∞, 0]nonnegativerational roots
• Veronese parent of A098172 → (θ+1)²
  1 · 0, 0, 9, 16 · 0, 0, 0, 0, 25, 72, 49 … Σ 1, 25, 146, 584, 1952, 5872, 16480, …
  (−∞, 0]nonnegative
• A039756 → x ↦ x+1 → D+1
  1 · 3, 1 · 12, 8, 1 · 62, 70, 19, 1 … Σ 1, 4, 21, 152, 1361, 13956, 159701, …
  (−∞, 0]nonnegative
• A038326: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
  1 · 11, 12 · 121, 264, 144 · 1331, 4356, 4752, 1728 … Σ 1, 23, 529, 12167, 279841, 6436343, …
  (−∞, 0)[−1, 0]nonnegative+1
• A062190 → reverse → x ↦ x/(x+1)
  1 · 6, 7 · 21, 56, 36 · 56, 252, 360, 165 … Σ 1, 13, 113, 833, 5641, 36365, 227305, …
  [−1, 0]nonnegative
• A137320 → basis A013609 → (θ+1)²
  1 · 2, 16 · 6, 80, 144 · 24, 416, 1296, 1024 … Σ 1, 18, 230, 2760, 33544, 424528, …
  (−∞, 0]nonnegative
• A125092 → θ+1 → x ↦ x+1
  1 · 9, 8 · 44, 70, 27 · 170, 378, 273, 64 … Σ 1, 17, 141, 885, 4761, 23193, 105381, …
  (−∞, −1]nonnegative
• A038221 → x ↦ x+1 → (θ+1)²
  1 · 6, 12 · 36, 144, 81 · 216, 1296, 1458, 432 … Σ 1, 18, 261, 3402, 41553, 485514, …
  (−∞, 0]nonnegative
• A001263 → basis A013609 → reverse
  1 · 2, 2 · 4, 10, 5 · 8, 36, 42, 14 … Σ 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, 0]nonnegative
• A063967 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 6, 22, 20 · 16, 88, 160, 96 … Σ 1, 6, 48, 360, 2736, 20736, 157248, …
  [−1, 0]nonnegative
• A105070 → basis A007318
  1 · 2 · 5, 2 · 12, 8 … Σ 1, 2, 7, 20, 61, 182, 547, 1640, …
  (−∞, −1]nonnegative
• A390433 → D
  1 · 4, 2 · 36, 40, 3 · 576, 1072, 168, 4 … Σ 0, 1, 6, 79, 1820, 64469, 3237138, …
  (−∞, 0]nonnegative
• A132393 → basis A013611 → reverse
  1 · 4, 1 · 16, 12, 2 · 64, 96, 44, 6 … Σ 1, 5, 30, 210, 1680, 15120, 151200, …
  (−∞, 0]nonnegativerational roots
• A166961 → reverse
  1 · 1, 1 · 1, 9, 1 · 1, 42, 59, 1 … Σ 1, 2, 11, 103, 1442, 28023, 712805, …
  (−∞, 0]nonnegative
• A034870 → Veronese 2 (j=1) → (θ+1)²
  2 · 4, 16 · 6, 80, 54 · 8, 224, 504, 128 … Σ 0, 2, 20, 140, 864, 4928, 26624, …
  (−∞, 0]nonnegative
• A104684 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 5 · 13, 42, 33 · 63, 303, 477, 245 … Σ 1, 8, 88, 1088, 14176, 190208, 2600704, …
  [−1, 0]nonnegative
• A142963 → reverse
  1 · 2, 1 · 4, 10, 1 · 8, 72, 30, 1 … Σ 1, 3, 15, 111, 1071, 12663, 177255, …
  (−∞, 0]nonnegative
• A111125 → basis A007318 → x ↦ x/(x+1)
  1 · 4, 5 · 11, 29, 19 · 29, 118, 159, 71 … Σ 1, 9, 59, 377, 2403, 15313, 97579, …
  [−1, −1/2)nonnegative
• A145324 → basis A013610 → basis A007318
  1 · 9, 6 · 117, 159, 54 · 1989, 4107, 2826, 648 … Σ 1, 15, 330, 9570, 344520, 14814360, …
  (−∞, −1]nonnegativerational roots
• A116412 → basis A013610 → x ↦ x+1
  1 · 7, 3 · 46, 42, 9 · 304, 423, 189, 27 … Σ 1, 10, 97, 943, 9166, 89095, 866017, …
  (−∞, −1]nonnegative
• A103327 → x ↦ x+1 → D+1
  1 · 5, 1 · 28, 14, 1 · 144, 128, 27, 1 … Σ 1, 6, 43, 300, 2037, 13554, 88799, …
  (−∞, 0]nonnegative
• A159854 → basis A008277 → x ↦ x+1
  5, 5, 1 · 20, 27, 10, 1 · 87, 148, 80, 16, 1 … Σ 11, 58, 332, 2048, 13528, 95160, …
  (−∞, −1]nonnegative
• A100862 → x ↦ x+1 → D
  1 · 5, 2 · 21, 18, 3 · 86, 114, 42, 4 … Σ 0, 1, 7, 42, 246, 1455, 8793, 54516, …
  (−∞, 0]nonnegative
• A038337 → basis A013609 → x ↦ x+1
  2025, 1980, 484 · 91125, 133650, 65340, 10648 … Σ 4489, 300763, 20151121, 1350125107, …
  (−∞, −1]nonnegativerational roots
• A104978: Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.
  1 · 1, 1 · 2, 5, 3 · 5, 21, 28, 12 … Σ 1, 2, 10, 66, 498, 4066, 34970, 312066, …
  (−∞, 0)nonnegative
• A159854 → Veronese 2 (j=1) → θ+1
  0, 2 · 0, 4 · 0, 6, 3 · 0, 8, 12 … Σ 0, 2, 4, 9, 20, 44, 96, 208, …
  (−∞, 0]nonnegative
• A080248 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 52, 90, 16 … Σ 1, 5, 26, 159, 1163, 10066, 101701, …
  (−∞, 0]nonnegative
• A133080 → θ+1 → D+1
  1 · 3, 2 · 0, 6, 3 · 0, 6, 15, 4 … Σ 1, 5, 9, 25, 25, 61, 49, 113, …
  (−∞, 0]nonnegative
• A274498 → x ↦ x+1 → θ+1
  1 · 3 · 9, 12 · 27, 36 … Σ 1, 3, 21, 63, 405, 1215, 7425, 22275, …
  (−∞, 0]nonnegativerational roots
• A225471 → basis A007318 → reverse
  1 · 1, 4 · 1, 12, 32 · 1, 24, 176, 384 … Σ 1, 5, 45, 585, 9945, 208845, 5221125, …
  (−∞, 0]nonnegativerational roots
• A124860: A Jacobsthal-Pascal triangle.
  1 · 1, 1 · 3, 6, 3 · 5, 15, 15, 5 … Σ 1, 2, 12, 40, 176, 672, 2752, 10880, …
  (−∞, 0)nonnegativerational roots
• A133080 → θ+1 → x ↦ x+1
  1 · 3, 2 · 3, 6, 3 · 7, 18, 15, 4 … Σ 1, 5, 12, 44, 80, 272, 448, 1472, …
  (−∞, −1]nonnegativerational roots
• A257608 → θ+1
  1 · 1, 2 · 1, 40, 3 · 1, 438, 657, 4 … Σ 1, 3, 44, 1100, 38280, 1697080, …
  (−∞, 0]nonnegative