Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• basis-map coefficients of A030528, shifted → basis A013609 → D
  -2 · 4, 8 · -6, -24, -24 · 8, 48, 96, 64 … Σ 0, -2, 12, -54, 216, -810, 2916, …
  (−∞, 0]rational roots
• A038220 → basis A007318 → x ↦ x/(x+1)
  1 · 5, 7 · 25, 70, 49 · 125, 525, 735, 343 … Σ 1, 12, 144, 1728, 20736, 248832, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A297191, shifted → θ+1 → D+1
  1 · 0, 6, 3 · 0, 0, 0, 20, 5 · 0, 0, 0, 0, 0, 42, 7 … Σ 1, 9, 25, 49, 81, 121, 169, 225, …
  (−∞, 0]nonnegativerational roots
• A127648 → Veronese 2 (j=1) → x ↦ x+1
  2 · 4, 4 · 6, 12, 6 · 8, 24, 24, 8 … Σ 0, 0, 2, 0, 8, 0, 24, 0, …
  (−∞, −1]nonnegativerational roots
• A110441 → basis A007318 → D+1
  1 · 5, 1 · 22, 10, 1 · 92, 68, 15, 1 … Σ 1, 6, 33, 176, 919, 4722, 23957, …
  (−∞, 0]nonnegative
• A038279 → basis A013610 → θ+1
  1 · 9, 6 · 81, 108, 27 · 729, 1458, 729, 108 … Σ 1, 15, 216, 3024, 41472, 559872, …
  (−∞, 0]nonnegativerational roots
• A169803 → basis A013609 → θ+1
  1 · 2, 4 · 3, 8 · 5, 20, 12 … Σ 1, 6, 11, 37, 84, 233, 565, 1458, …
  (−∞, 0]nonnegative
• A038220 → basis A013611 → (θ+1)²
  1 · 5, 32 · 25, 320, 576 · 125, 2400, 8640, 8192 … Σ 1, 37, 921, 19357, 369265, 6610773, …
  (−∞, 0]nonnegative
• A128813 → basis A013610 → (θ+1)²
  1 · 2, 12 · 8, 120, 243 · 56, 1416, 6561, 7776 … Σ 1, 14, 371, 15809, 990619, 85875934, …
  (−∞, 0]nonnegative
• A082985 → (θ+1)²
  1 · 1, 12 · 1, 20, 45 · 1, 28, 126, 112 … Σ 1, 13, 66, 267, 985, 3444, 11597, …
  (−∞, 0]nonnegative
• A178820 → basis A013610 → x ↦ x+1
  1 · 20, 12 · 250, 300, 90 · 2500, 4500, 2700, 540 … Σ 1, 32, 640, 10240, 143360, 1835008, …
  (−∞, −1]nonnegativerational roots
• A359363 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 2, 12, 4 … Σ 1, 2, 5, 18, 77, 368, 1899, 10370, …
  (−∞, 0]nonnegative
• A013609 → basis A008288 → D
  2 · 16, 8 · 82, 104, 24 · 352, 784, 432, 64 … Σ 0, 2, 24, 210, 1632, 11890, 83160, …
  (−∞, 0]nonnegative
• Hadamard-factor parent of A127773 → D
  3 · 0, 24 · 0, 0, 180 · 0, 0, 0, 1440 … Σ 0, 3, 24, 180, 1440, 12600, 120960, …
  (−∞, 0]nonnegativerational roots
• A185263 → basis A007318
  1 · 1 · 2, 1 · 6, 5 … Σ 1, 1, 3, 11, 63, 407, 3457, 32081, …
  (−∞, −1]nonnegative
• A111577 → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 35, 48, 15, 1 … Σ 1, 3, 15, 99, 783, 7155, 73791, 843939, …
  (−∞, −1]nonnegative
• A347823 → (θ+1)²
  1 · 2, 12 · 3, 32, 45 · 4, 60, 162, 112 … Σ 1, 14, 80, 338, 1216, 3952, 11968, …
  (−∞, 0]nonnegative
• Veronese parent of A109446 → Veronese 2 (j=1) → D
  1 · 1 · 6, 2 · 10, 2 … Σ 0, 0, 1, 1, 8, 12, 48, 80, …
  (−∞, 0]nonnegative
• A136124 → basis A013610 → x ↦ x+1
  1 · 6, 3 · 42, 39, 9 · 336, 438, 189, 27 … Σ 1, 9, 90, 990, 11880, 154440, 2162160, …
  (−∞, −1]nonnegativerational roots
• A204420 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 6, 9 · 0, 120, 330, 225 … Σ 1, 1, 15, 675, 61425, 9398025, …
  [−1, 0]nonnegativerational roots
• A038322 → basis A013609 → x ↦ x/(x+1)
  1 · 19, 35 · 361, 1330, 1225 … Σ 1, 54, 2916, 157464, 8503056, …
  [−1, 0]nonnegativerational roots
• A048993 → θ+1 → D
  2 · 2, 6 · 2, 18, 12 · 2, 42, 72, 20 … Σ 0, 2, 8, 32, 136, 622, 3060, 16130, …
  (−∞, 0]nonnegative
• A269951 → basis A007318
  1 · 1, 1 · 3, 4, 1 · 11, 18, 8, 1 … Σ 1, 2, 8, 38, 212, 1370, 10112, 84158, …
  (−∞, −1]nonnegative
• A304236 → basis A013609 → D+1
  1 · 1 · 10, 6 · 19, 12 … Σ 1, 1, 16, 31, 235, 628, 3397, 10729, …
  (−∞, 0]nonnegative
• A257622 → θ+1
  1 · 4, 8 · 16, 112, 48 · 64, 1104, 1656, 256 … Σ 1, 12, 176, 3080, 62832, 1466080, …
  (−∞, 0]nonnegative
• A272866 → basis A013611 → reverse
  1 · 16, 20, 5 · 256, 640, 560, 200, 25 … Σ 1, 41, 1681, 68921, 2825761, 115856201, …
  (−∞, 0]nonnegative
• A137597 → x ↦ x+1 → D+1
  1 · 4, 1 · 18, 9, 1 · 90, 67, 15, 1 … Σ 1, 5, 28, 173, 1165, 8468, 65923, …
  (−∞, −1]nonnegative
• A108838 → D → x ↦ x+1
  2 · 12, 4 · 56, 42, 6 · 240, 288, 96, 8 … Σ 0, 2, 16, 104, 632, 3730, 21696, …
  (−∞, −1]nonnegative
• A225466 → D+1
  1 · 5, 3 · 25, 39, 9 · 125, 387, 216, 27 … Σ 1, 8, 73, 755, 8740, 111761, 1560817, …
  (−∞, 0]nonnegative
• A062145 → reverse → D
  1 · 10, 2 · 45, 36, 3 · 140, 252, 84, 4 … Σ 0, 1, 12, 84, 480, 2475, 12012, 56056, …
  (−∞, 0]nonnegative
• Hadamard-factor parent of A127773 → (θ+1)²
  1 · 0, 12 · 0, 0, 108 · 0, 0, 0, 960 … Σ 1, 12, 108, 960, 9000, 90720, 987840, …
  (−∞, 0]nonnegativerational roots
• A137320 → x ↦ x+1 → D+1
  1 · 4, 2 · 16, 18, 4 · 76, 124, 60, 8 … Σ 1, 6, 38, 268, 2116, 18608, 180912, …
  (−∞, 0]nonnegative
• A027466 → basis A013611
  1 · 8, 4 · 64, 64, 16 · 512, 768, 384, 64 … Σ 1, 12, 144, 1728, 20736, 248832, …
  (−∞, −1]nonnegativerational roots
• A038338 → basis A013609 → D
  1152, 1152 · 41472, 82944, 41472 … Σ 2304, 165888, 10616832, 637009920, …
  (−∞, 0]nonnegativerational roots
• A132393 → basis A013609 → x ↦ x/(x+1)
  1 · 1, 3 · 2, 10, 12 · 6, 40, 86, 60 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A026729, shifted → basis A013610 → (θ+1)²
  1 · -1, -12 · 1, 24, 81 · -1, -36, -243, -432 … Σ 1, -13, 106, -712, 4288, -24064, …
  (−∞, 0]
• A013613 → x ↦ x+1
  1 · 7, 6 · 49, 84, 36 · 343, 882, 756, 216 … Σ 1, 13, 169, 2197, 28561, 371293, …
  (−∞, −1]nonnegativerational roots
• A356546 → x ↦ x+1 → D+1
  1 · 6, 2 · 48, 36, 6 · 400, 480, 180, 20 … Σ 1, 8, 90, 1080, 13230, 163296, 2020788, …
  (−∞, 0]nonnegativerational roots
• A130850 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 6, 30, 49, 26 … Σ 1, 3, 15, 111, 1095, 13503, 199815, …
  [−1, 0]nonnegative
• A110441 → basis A007318 → x ↦ x/(x+1)
  1 · 4, 5 · 14, 36, 23 · 48, 188, 244, 105 … Σ 1, 9, 73, 585, 4681, 37449, 299593, …
  [−1, −1/2)nonnegative
• A207824 → θ+1
  1 · 5, 2 · 24, 20, 3 · 115, 146, 45, 4 … Σ 1, 7, 47, 310, 2017, 12981, 82799, …
  (−∞, 0]nonnegative
• A130123 → basis A013611 → D
  8 · 32, 128 · 96, 768, 1536 · 256, 3072, 12288, 16384 … Σ 0, 8, 160, 2400, 32000, 400000, …
  (−∞, 0]nonnegativerational roots
• A063967 → reverse → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 6 · 1, 8, 20, 16 … Σ 1, 3, 12, 45, 171, 648, 2457, 9315, …
  [−1, 0]nonnegative
• A132393 → Veronese 2 → D+1
  1 · 1, 1 · 3, 3 · 11, 13, 1 … Σ 1, 0, 2, 6, 25, 130, 807, 5817, …
  (−∞, 0]nonnegative
• A038207 → Veronese 2 (j=1) → D
  1 · 8 · 40, 2 · 160, 24 … Σ 0, 0, 0, 1, 8, 42, 184, 731, …
  (−∞, 0]nonnegative
• A090452 → x ↦ x/(x+1)
  1 · 1, 5, 6 · 1, 11, 43, 72, 44 … Σ 1, 12, 171, 2700, 45522, 802872, …
  [−1, 0]nonnegative
• A081578 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 7, 28, 9 · 20, 120, 108, 16 … Σ 1, 6, 44, 264, 1548, 8640, 46980, …
  (−∞, 0]nonnegative
• A128813 → (θ+1)²
  1 · 1, 4 · 1, 16, 27 · 1, 40, 243, 288 … Σ 1, 5, 44, 572, 10332, 248772, 7730568, …
  (−∞, 0]nonnegative
• A035342 → D
  1 · 9, 2 · 87, 36, 3 · 975, 570, 90, 4 … Σ 0, 1, 11, 126, 1639, 24305, 407196, …
  (−∞, 0]nonnegative
• A121574 → x ↦ x+1 → reverse
  1 · 1, 3 · 1, 7, 10 · 1, 11, 35, 33 … Σ 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  (−∞, 0]nonnegative