Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A286718 → basis A013609 → D
  2 · 14, 8 · 132, 120, 24 · 1612, 1848, 624, 64 … Σ 0, 2, 22, 276, 4148, 73648, 1515792, …
  (−∞, 0]nonnegative
• A055151 → D+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 19, 51, 141, 393, …
  (−∞, 0]nonnegative
• A081582 → x ↦ x+1
  1 · 2, 1 · 11, 11, 1 · 36, 54, 20, 1 … Σ 1, 3, 23, 111, 655, 3519, 19727, …
  (−∞, −1]nonnegative
• A249250 → (θ+1)²
  1 · 1, 12 · 1, 24, 27 · 1, 36, 135, 48 … Σ 1, 13, 52, 220, 832, 3028, 10600, …
  (−∞, 0]nonnegative
• A225470 → basis A013611 → x ↦ x/(x+1)
  1 · 3, 7 · 18, 72, 70 · 162, 882, 1566, 910 … Σ 1, 10, 160, 3520, 98560, 3351040, …
  [−1, 0]nonnegativerational roots
• A121314 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 2, 9, 4 … Σ 1, 2, 5, 15, 46, 139, 413, 1210, …
  (−∞, 0]nonnegative
• A124932 → reverse → θ+1
  1 · 3, 4 · 6, 18, 9 · 10, 48, 54, 16 … Σ 1, 7, 33, 128, 440, 1392, 4144, 11776, …
  (−∞, 0]nonnegative
• A038281 → D+1
  112, 66, 9 · 1088, 1008, 297, 27 … Σ 187, 2420, 30613, 380666, 4670479, …
  (−∞, 0]nonnegativerational roots
• A103327 → basis A013611 → θ+1
  1 · 4, 8 · 16, 96, 48 · 64, 640, 1152, 256 … Σ 1, 12, 160, 2112, 26880, 331776, …
  (−∞, 0]nonnegative
• A116412 → basis A013610 → θ+1
  1 · 4, 6 · 13, 48, 27 · 43, 252, 324, 108 … Σ 1, 10, 88, 727, 5764, 44428, 335440, …
  (−∞, 0]nonnegative
• A003506 → (θ+1)² → x ↦ x+1
  1 · 10, 8 · 54, 78, 27 · 224, 456, 300, 64 … Σ 0, 1, 18, 159, 1044, 5805, 28998, …
  (−∞, −1]nonnegative
• A133080 → θ+1
  1 · 1, 2 · 0, 0, 3 · 0, 0, 3, 4 … Σ 1, 3, 3, 7, 5, 11, 7, 15, …
  (−∞, 0]nonnegativerational roots
• A088996 → basis A013611 → (θ+1)²
  1 · 1, 16 · 3, 80, 288 · 15, 544, 3600, 6144 … Σ 1, 17, 371, 10303, 348665, 13920945, …
  (−∞, 0]nonnegative
• A257618 → x ↦ x+1
  1 · 4, 2 · 48, 48, 4 · 960, 1440, 496, 8 … Σ 1, 6, 100, 2904, 117520, 6105696, …
  (−∞, −1]nonnegative
• A103327 → reverse → D
  3 · 10, 10 · 21, 70, 21 · 36, 252, 252, 36 … Σ 0, 3, 20, 112, 576, 2816, 13312, 61440, …
  (−∞, 0]nonnegative
• A104029: Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).
  1 · 2, 1 · 3, 5, 1 · 4, 13, 9, 1 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  (−∞, 0)nonnegative
• A028326 → basis A007318 → (θ+1)²
  2 · 4, 8 · 8, 32, 18 · 16, 96, 108, 32 … Σ 2, 12, 58, 252, 1026, 3996, 15066, …
  (−∞, 0]nonnegative
• A178821 → basis A013611 → D
  20 · 240, 480 · 1680, 6720, 6720 … Σ 0, 20, 720, 15120, 241920, 3265920, …
  (−∞, 0]nonnegativerational roots
• A126351 → (θ+1)²
  1 · 1, 8 · 1, 20, 36 · 1, 36, 171, 128 … Σ 1, 9, 57, 336, 1992, 12193, 77725, …
  (−∞, 0]nonnegative
• A013620 → basis A013610 → D+1
  1 · 14, 9 · 115, 252, 81 · 800, 3105, 3402, 729 … Σ 1, 23, 448, 8036, 137200, 2266544, …
  (−∞, 0]nonnegativerational roots
• A128813 → basis A013610 → θ+1
  1 · 2, 6 · 8, 60, 81 · 56, 708, 2187, 1944 … Σ 1, 8, 149, 4895, 249445, 18214270, …
  (−∞, 0]nonnegative
• A269951 → (θ+1)²
  1 · 0, 4 · 0, 8, 9 · 0, 20, 45, 16 … Σ 1, 4, 17, 81, 440, 2730, 19239, 152675, …
  (−∞, 0]nonnegative
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 8, 12 · 36, 108, 81 · 128, 576, 864, 432 … Σ 1, 20, 225, 2000, 15625, 112500, …
  [−1, 0]nonnegativerational roots
• A008288 → basis A007318 → D+1
  1 · 3, 1 · 10, 7, 1 · 30, 34, 11, 1 … Σ 1, 4, 18, 76, 314, 1272, 5078, 20036, …
  (−∞, 0]nonnegative
• A304236 → basis A013610 → D
  9 · 18 · 81, 162 · 198, 486 … Σ 0, 0, 9, 18, 243, 684, 5229, 17766, …
  (−∞, 0]nonnegative
• A081577 → basis A013610 → D+1
  1 · 5, 3 · 24, 36, 9 · 88, 252, 171, 27 … Σ 1, 8, 69, 538, 4055, 29676, 212809, …
  (−∞, 0]nonnegative
• A344565 → D+1
  3 · 30, 15 · 135, 180, 45 · 420, 945, 630, 105 … Σ 3, 45, 360, 2100, 10080, 42336, 161280, …
  (−∞, 0]nonnegativerational roots
• A286718 → basis A013611 → (θ+1)²
  1 · 2, 16 · 10, 112, 144 · 80, 1056, 2160, 1024 … Σ 1, 18, 266, 4320, 80064, 1690848, …
  (−∞, 0]nonnegative
• A038314 → basis A013609 → θ+1
  484, 2112, 1728 · 10648, 69696, 114048, 55296 … Σ 4324, 249688, 13821712, 743257696, …
  (−∞, 0]nonnegativerational roots
• A013622 → basis A013610 → x ↦ x/(x+1)
  1 · 8, 23 · 64, 368, 529 · 512, 4416, 12696, 12167 … Σ 1, 31, 961, 29791, 923521, 28629151, …
  [−1, 0]nonnegativerational roots
• A038291 → basis A013609 → basis A013610
  1 · 12, 6 · 144, 144, 36 · 1728, 2592, 1296, 216 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, −1]nonnegativerational roots
• A038331 → basis A013609 → θ+1
  289, 680, 300 · 4913, 17340, 15300, 4000 … Σ 1269, 41553, 1318761, 40920957, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729 → basis A013611 → x ↦ x+1
  1 · 5, 4 · 25, 40, 16 · 125, 300, 240, 64 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1]nonnegativerational roots
• A133214 → reverse
  4, 8, 1 · 8, 36, 18, 1 · 16, 128, 144, 32, 1 … Σ 13, 63, 321, 1683, 8989, 48639, 265729, …
  (−∞, 0]nonnegative
• A139548 → x ↦ x+1 → basis A007318
  1 · 18, 12, 2 · 324, 432, 216, 48, 4 … Σ 1, 32, 1024, 32768, 1048576, 33554432, …
  (−∞, −1]nonnegativerational roots
• A081579 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 8, 32, 9 · 24, 144, 126, 16 … Σ 1, 6, 49, 310, 1965, 11742, 68657, …
  (−∞, 0]nonnegative
• A391045 → D+1
  1 · 2, 1 · 5, 6, 1 · 10, 29, 13, 1 … Σ 1, 3, 12, 53, 244, 1152, 5531, 26875, …
  (−∞, 0]nonnegative
• A114655 → θ+1 → D+1
  2 · 8, 4 · 32, 36, 6 · 112, 240, 96, 8 … Σ 2, 12, 74, 456, 2802, 17156, 104666, …
  (−∞, 0]nonnegative
• A257620 → D
  3 · 36, 18 · 297, 594, 81 · 2106, 10692, 6318, 324 … Σ 0, 3, 54, 972, 19440, 437400, 11022480, …
  (−∞, 0]nonnegative
• A013611 → basis A013611 → (θ+1)²
  1 · 5, 64 · 25, 640, 2304 · 125, 4800, 34560, 65536 … Σ 1, 69, 2969, 105021, 3327345, 98175861, …
  (−∞, 0]nonnegative
• A257609 → θ+1
  1 · 2, 4 · 4, 32, 12 · 8, 176, 264, 32 … Σ 1, 6, 48, 480, 5760, 80640, 1290240, …
  (−∞, 0]nonnegative
• A225471 → basis A013611 → D+1
  1 · 8, 4 · 80, 80, 16 · 1088, 1472, 576, 64 … Σ 1, 12, 176, 3200, 70144, 1806336, …
  (−∞, 0]nonnegative
• A126351 → x ↦ x+1 → θ+1
  1 · 3, 4 · 10, 26, 12 · 37, 142, 129, 32 … Σ 1, 7, 48, 340, 2551, 20473, 175954, …
  (−∞, 0]nonnegative
• A178822 → basis A013611 → x ↦ x/(x+1)
  1 · 12, 36 · 84, 504, 756 · 448, 4032, 12096, 12096 … Σ 1, 48, 1344, 28672, 516096, 8257536, …
  [−1, 0]nonnegativerational roots
• A152198 → basis A013611 → basis A007318
  1 · 1 · 6, 4 · 6, 4 … Σ 1, 1, 10, 10, 100, 100, 1000, 1000, …
  (−∞, −1]nonnegativerational roots
• A257617 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 44, 44 · 8, 412, 1188, 792 … Σ 1, 6, 92, 2400, 86696, 4003656, …
  [−1, 0]nonnegative
• A100862 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … Σ 1, 5, 22, 95, 415, 1847, 8410, 39209, …
  (−∞, 0]nonnegative
• A225468 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 15, 12 · 8, 63, 117, 63 … Σ 1, 5, 31, 251, 2569, 31181, 430471, …
  [−1, 0]nonnegative
• A308737 → reverse
  1 · 3, 1 · 7, 8, 1 · 15, 31, 17, 1 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0]nonnegativerational roots
• A156992 → basis A013609 → basis A007318
  1 · 8, 4 · 96, 96, 24 · 1536, 2304, 1152, 192 … Σ 1, 12, 216, 5184, 155520, 5598720, …
  (−∞, −1]nonnegativerational roots