Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A028421 → θ+1 → reverse
  1 · 4, 1 · 9, 12, 2 · 16, 54, 44, 6 … Σ 1, 5, 23, 120, 724, 5012, 39332, …
  (−∞, 0]nonnegative
• A137320 → basis A013609 → θ+1
  1 · 2, 8 · 6, 40, 48 · 24, 208, 432, 256 … Σ 1, 10, 94, 920, 9624, 108336, 1312656, …
  (−∞, 0]nonnegative
• A038277 → basis A013609 → reverse
  1 · 22, 18 · 484, 792, 324 … Σ 1, 40, 1600, 64000, 2560000, 102400000, …
  (−∞, 0]nonnegativerational roots
• A100861 → basis A007318
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 25, 81, 331, 1303, …
  (−∞, −1]nonnegative
• A196347 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 8, 48, 72 · 48, 432, 1296, 1296 … Σ 1, 8, 128, 3072, 98304, 3932160, …
  [−1, 0]nonnegativerational roots
• A085880 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 32, 32, 8 · 320, 480, 240, 40 … Σ 1, 6, 72, 1080, 18144, 326592, 6158592, …
  (−∞, −1]nonnegativerational roots
• A120910 → Veronese 2 → θ+1
  3 · 6 · 12, 6 · 24, 36 … Σ 3, 6, 18, 60, 201, 666, 2184, 7104, …
  (−∞, 0]nonnegative
• A062190 → reverse
  1 · 6, 1 · 21, 14, 1 · 56, 84, 24, 1 … Σ 1, 7, 36, 165, 715, 3003, 12376, 50388, …
  (−∞, 0]nonnegative
• A132774 → D+1
  1 · 5, 3 · 4, 14, 5 · 0, 12, 27, 7 … Σ 1, 8, 23, 46, 77, 116, 163, 218, …
  (−∞, 0]nonnegative
• A219512 → reverse
  1 · 3, 1 · 15, 12, 1 · 105, 141, 33, 1 … Σ 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0]nonnegative
• A127648 → (θ+1)² → (θ+1)²
  1 · 0, 32 · 0, 0, 243 · 0, 0, 0, 1024 … Σ 0, 1, 32, 243, 1024, 3125, 7776, 16807, …
  (−∞, 0]nonnegativerational roots
• A008459 → basis A013611 → (θ+1)²
  1 · 2, 16 · 6, 96, 144 · 20, 480, 1728, 1024 … Σ 1, 18, 246, 3252, 42150, 534876, …
  (−∞, 0]nonnegative
• A290319 → basis A013609 → D+1
  1 · 4, 2 · 28, 24, 4 · 304, 328, 96, 8 … Σ 1, 6, 56, 736, 12544, 263168, 6561792, …
  (−∞, 0]nonnegative
• A178822 → basis A013610 → D
  18 · 252, 378 · 2016, 6048, 4536 … Σ 0, 18, 630, 12600, 189000, 2362500, …
  (−∞, 0]nonnegativerational roots
• A114656 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 5 · 11, 38, 31 · 45, 237, 399, 215 … Σ 1, 8, 80, 896, 10752, 135168, 1757184, …
  [−1, 0]nonnegative
• A249250 → x ↦ x+1 → θ+1
  1 · 4, 6 · 10, 24, 9 · 28, 96, 72, 12 … Σ 1, 10, 43, 208, 949, 4294, 19135, …
  (−∞, 0]nonnegative
• A185410 → D+1
  1 · 1 · 3, 2 · 11, 18, 4 … Σ 1, 1, 5, 33, 285, 3045, 38745, 571725, …
  (−∞, 0]nonnegative
• A128099 → basis A013611 → x ↦ x+1
  1 · 1 · 11, 8 · 21, 16 … Σ 1, 1, 19, 37, 379, 1045, 7867, 26677, …
  (−∞, −1]nonnegative
• chebyshev_u → x ↦ x+1 → (θ+1)²
  1 · 2, 8 · 3, 32, 36 · 4, 80, 216, 128 … Σ 1, 10, 71, 428, 2345, 12062, 59311, …
  (−∞, 0]nonnegative
• A178822 → Veronese 2 → D+1
  1 · 6 · 42, 21 · 224, 168 … Σ 1, 6, 63, 392, 2016, 9072, 36960, …
  (−∞, 0]nonnegative
• A166344 → (θ+1)²
  1, 24, 9 · 1, 60, 135, 16 · 1, 136, 810, 544, 25 … Σ 34, 212, 1516, 12304, 111920, 1128480, …
  (−∞, 0]nonnegative
• A131436 → θ+1
  1 · 0, 6 · 0, 0, 21 · 0, 0, 0, 60 … Σ 1, 6, 21, 60, 155, 378, 889, 2040, …
  (−∞, 0]nonnegativerational roots
• A038337 → x ↦ x/(x+1)
  144, 552, 529 · 1728, 9936, 19044, 12167 … Σ 1225, 42875, 1500625, 52521875, …
  [−1, 0]nonnegativerational roots
• A181543 → D+1
  1 · 2, 1 · 9, 10, 1 · 28, 81, 30, 1 … Σ 1, 3, 20, 140, 1038, 7882, 60736, …
  (−∞, 0]nonnegative
• A023531 → D+1
  1 · 1, 1 · 0, 2, 1 · 0, 0, 3, 1 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0]nonnegativerational roots
• A091044 → basis A013609 → (θ+1)²
  1 · 4, 16 · 16, 128, 108 · 64, 768, 1440, 512 … Σ 1, 20, 252, 2784, 28688, 282048, …
  (−∞, 0]nonnegative
• A027465 → Veronese 2 (j=1) → x ↦ x+1
  1 · 6 · 28, 1 · 120, 12 … Σ 0, 1, 6, 29, 132, 589, 2610, 11537, …
  (−∞, −1]nonnegative
• A038327: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
  1 · 12, 1 · 144, 24, 1 · 1728, 432, 36, 1 … Σ 1, 13, 169, 2197, 28561, 371293, …
  (−∞, −1]nonnegativerational roots
• A156365 → θ+1
  1, 16, 12 · 1, 44, 132, 32 · 1, 104, 792, 832, 80 … Σ 29, 209, 1809, 18269, 210853, 2737769, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → D+1 → x ↦ x/(x+1)
  1 · 3, 5 · 7, 26, 22 · 15, 95, 175, 99 … Σ 1, 8, 55, 384, 2841, 22504, 190863, …
  [−1, 0]nonnegative
• A059473 → (θ+1)²
  1 · 2, 8 · 4, 40, 36 · 8, 128, 288, 128 … Σ 1, 10, 80, 552, 3508, 21080, 121664, …
  (−∞, 0]nonnegative
• A030528 → x ↦ x−3
  1 · -2, 1 · 3, -4, 1 · -3, 10, -6, 1 … Σ 1, -1, 0, 2, -4, 4, 0, -8, …
  (−1, 3]
• A028326 → basis A013610 → θ+1
  2 · 4, 12 · 8, 48, 54 · 16, 144, 324, 216 … Σ 2, 16, 110, 700, 4250, 25000, 143750, …
  (−∞, 0]nonnegativerational roots
• A143685 → basis A007318 → reverse
  1 · 1, 2 · 1, 13, 13 · 1, 24, 66, 44 … Σ 1, 3, 27, 135, 891, 5103, 31347, …
  (−∞, 0]nonnegative
• A219512 → θ+1
  1 · 1, 6 · 1, 24, 45 · 1, 66, 423, 420 … Σ 1, 7, 70, 910, 14560, 276640, 6086080, …
  (−∞, 0]nonnegative
• A143491 → basis A013611 → x ↦ x/(x+1)
  12, 52, 56 · 60, 368, 748, 504 … Σ 120, 1680, 26880, 483840, 9676800, …
  [−1, 0]nonnegativerational roots
• A027467 → basis A013611 → x ↦ x/(x+1)
  1 · 16, 20 · 256, 640, 400 · 4096, 15360, 19200, 8000 … Σ 1, 36, 1296, 46656, 1679616, 60466176, …
  [−1, 0]nonnegativerational roots
• A208509 → (θ+1)²
  1 · 3 · 5, 4 · 7, 20 … Σ 1, 3, 9, 27, 74, 194, 492, 1216, …
  (−∞, 0]nonnegative
• A125662 → basis A013611 → (θ+1)²
  1 · 4, 16 · 15, 128, 144 · 56, 736, 1728, 1024 … Σ 1, 20, 287, 3544, 40097, 428092, …
  (−∞, 0]nonnegative
• A132393 → D → θ+1
  1 · 1, 4 · 2, 12, 9 · 6, 44, 54, 16 … Σ 0, 1, 5, 23, 120, 724, 5012, 39332, …
  (−∞, 0]nonnegative
• A059364 → (θ+1)²
  1 · 2, 4 · 6, 28, 18 · 24, 184, 261, 96 … Σ 1, 6, 52, 565, 7321, 109886, 1872704, …
  (−∞, 0]nonnegative
• A225470: Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
  1 · 2, 1 · 10, 7, 1 · 80, 66, 15, 1 … Σ 1, 3, 18, 162, 1944, 29160, 524880, …
  (−∞, −1]nonnegativerational roots
• A100861 → basis A007318 → D+1
  1 · 1 · 3, 1 · 7, 3 … Σ 1, 1, 4, 10, 43, 151, 706, 3004, …
  (−∞, 0]nonnegative
• A207824 → x ↦ x+1 → D
  1 · 12, 2 · 106, 36, 3 · 828, 426, 72, 4 … Σ 0, 1, 14, 145, 1330, 11420, 94066, …
  (−∞, 0]nonnegative
• A001263 → basis A013609 → D+1
  1 · 4, 2 · 15, 18, 4 · 56, 114, 60, 8 … Σ 1, 6, 37, 238, 1578, 10684, 73429, …
  (−∞, 0]nonnegative
• A166973 → reverse
  1 · 1, 1 · 1, 7, 1 · 1, 18, 43, 1 … Σ 1, 2, 9, 63, 536, 5307, 60389, 775988, …
  (−∞, 0]nonnegative
• A272866 → basis A013609 → x ↦ x/(x+1)
  1 · 5, 20, 19 · 25, 200, 590, 760, 361 … Σ 1, 44, 1936, 85184, 3748096, 164916224, …
  [−1, 0]nonnegative
• A225472 → θ+1
  1 · 2, 6 · 4, 42, 54 · 8, 234, 810, 648 … Σ 1, 8, 100, 1700, 36388, 937988, …
  (−∞, 0]nonnegative
• A135065 → x ↦ x+1 → D
  4 · 36, 18 · 192, 192, 48 · 800, 1200, 600, 100 … Σ 0, 4, 54, 432, 2700, 14580, 71442, …
  (−∞, 0]nonnegativerational roots
• A008277 → θ+1 → D+1 → D+1
  1 · 5, 2 · 19, 18, 3 · 65, 110, 42, 4 … Σ 1, 7, 40, 221, 1241, 7218, 43807, …
  (−∞, 0]nonnegative