Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A099040 → θ+1 → D+1
  1 · 4, 4 · 4, 28, 12 · 0, 48, 120, 32 … Σ 1, 8, 44, 200, 820, 3136, 11424, 40128, …
  (−∞, 0]nonnegative
• A122848 → basis A007318 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 4, 21, 36, 20 … Σ 1, 3, 15, 81, 513, 3483, 25839, 202905, …
  [−1, 0]nonnegative
• A038330 → basis A013611 → D
  512, 512 · 12288, 24576, 12288 … Σ 1024, 49152, 2097152, 83886080, …
  (−∞, 0]nonnegativerational roots
• A347056 → D+1
  1 · 7, 4 · 22, 36, 10 · 50, 140, 110, 20 … Σ 1, 11, 68, 320, 1280, 4592, 15232, …
  (−∞, 0]nonnegative
• A137320 → x ↦ x+1 → (θ+1)²
  1 · 2, 8 · 6, 40, 36 · 24, 208, 324, 128 … Σ 1, 10, 82, 684, 6100, 58944, 618368, …
  (−∞, 0]nonnegative
• Veronese parent of A098172 → basis A007318
  1 · 2, 5, 4, 1 · 4, 20, 41, 44, 26, 8, 1 … Σ 1, 12, 144, 1728, 20736, 248832, …
  (−∞, −1]nonnegativerational roots
• A347056 → x ↦ x+1
  1 · 7, 4 · 32, 36, 10 · 120, 200, 110, 20 … Σ 1, 11, 78, 450, 2295, 10773, 47628, …
  (−∞, −1]nonnegativerational roots
• A038334 → basis A013610 → reverse
  576, 960, 400 · 13824, 34560, 28800, 8000 … Σ 1936, 85184, 3748096, 164916224, …
  (−∞, 0]nonnegativerational roots
• A114687 → reverse → D
  1 · 6, 2 · 24, 24, 3 · 80, 160, 60, 4 … Σ 0, 1, 8, 51, 304, 1765, 10128, 57799, …
  (−∞, 0]nonnegative
• A201637 → (θ+1)²
  1 · 1 · 1, 8 · 1, 32, 54 … Σ 1, 1, 9, 87, 995, 13265, 202545, …
  (−∞, 0]nonnegative
• A132014: T(n,j) for double application of an iterated mixed order Laguerre transform: Coefficients of Laguerre polynomial (-1)^n*n!*L(n,2-n,x).
  A023531 → D²−2D+1 (shift -1)
  1 · -2, 1 · 2, -4, 1 · 0, 6, -6, 1 … Σ 1, -1, -1, 1, 5, 11, 19, 29, …
  (−∞, ∞)
• A225471 → basis A013609 → D
  2 · 24, 8 · 352, 192, 24 · 6400, 4480, 960, 64 … Σ 0, 2, 32, 568, 11904, 292128, 8260608, …
  (−∞, 0]nonnegative
• A114656 → basis A013610 → θ+1
  1 · 3, 6 · 11, 48, 27 · 45, 306, 405, 108 … Σ 1, 9, 86, 864, 8960, 94896, 1020064, …
  (−∞, 0]nonnegative
• A304236 → basis A013610 → θ+1
  1 · 1 · 4, 18 · 7, 36 … Σ 1, 1, 22, 43, 424, 1165, 7882, 26191, …
  (−∞, 0]nonnegative
• A132393 → basis A008277 → D
  1 · 4, 2 · 18, 18, 3 · 96, 144, 48, 4 … Σ 0, 1, 6, 39, 292, 2505, 24306, 263431, …
  (−∞, 0]nonnegative
• A085478 → basis A013611 → D
  4 · 20, 32 · 76, 256, 192 · 260, 1344, 2112, 1024 … Σ 0, 0, 4, 52, 524, 4740, 40360, 330584, …
  (−∞, 0]nonnegative
• A142461 → D
  1 · 14, 2 · 111, 222, 3 · 796, 5772, 2388, 4 … Σ 0, 1, 16, 336, 8960, 291200, 11182080, …
  (−∞, 0]nonnegative
• A104684 → basis A013610 → x ↦ x+1
  1 · 6, 3 · 46, 42, 9 · 396, 522, 216, 27 … Σ 1, 9, 97, 1161, 14721, 192969, 2582881, …
  (−∞, −1]nonnegative
• A304236 → basis A013610 → x ↦ x+1
  1 · 1 · 13, 9 · 25, 18 … Σ 1, 1, 22, 43, 505, 1408, 12013, 41581, …
  (−∞, −1]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → (θ+1)²
  1 · 2 · 4, 8 · 8, 64 … Σ 1, 2, 12, 72, 512, 4144, 37632, 379008, …
  (−∞, 0]nonnegative
• A124932 → reverse → D+1
  1 · 5, 2 · 15, 15, 3 · 34, 60, 30, 4 … Σ 1, 7, 33, 128, 440, 1392, 4144, 11776, …
  (−∞, 0]nonnegative
• A178822 → Veronese 2 → (θ+1)²
  1 · 6 · 21, 84 · 56, 672 … Σ 1, 6, 105, 728, 4284, 21672, 97944, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → Veronese 2 → D
  2 · 0, 48 · 0, 0, 2160 · 0, 0, 0, 161280 … Σ 0, 0, 2, 0, 48, 0, 2160, 0, …
  (−∞, 0]nonnegativerational roots
• A136572 → basis A013611 → θ+1
  1 · 1, 8 · 2, 32, 96 · 6, 144, 864, 1536 … Σ 1, 9, 130, 2550, 63000, 1875000, …
  (−∞, 0]nonnegativerational roots
• A038284 → basis A007318 → D
  168, 72 · 3528, 3024, 648 · 65856, 84672, 36288, 5184 … Σ 240, 7200, 192000, 4800000, 115200000, …
  (−∞, 0]nonnegativerational roots
• A121574 → basis A013611 → D
  4 · 28, 32 · 140, 352, 192 · 620, 2432, 2880, 1024 … Σ 0, 4, 60, 684, 6956, 66408, 609048, …
  (−∞, 0]nonnegative
• A027467 → x ↦ x/(x+1)
  1 · 15, 16 · 225, 480, 256 · 3375, 10800, 11520, 4096 … Σ 1, 31, 961, 29791, 923521, 28629151, …
  [−1, 0]nonnegativerational roots
• A013613 → basis A013611 → x ↦ x/(x+1)
  1 · 7, 31 · 49, 434, 961 · 343, 4557, 20181, 29791 … Σ 1, 38, 1444, 54872, 2085136, 79235168, …
  [−1, 0]nonnegativerational roots
• A272867 → D
  4, 2 · 8, 36, 24, 4 · 12, 102, 264, 204, 60, 6 … Σ 0, 6, 72, 648, 5184, 38880, 279936, …
  (−∞, 0]nonnegative
• A008556 → x ↦ x+1
  1 · 2 · 8, 2 · 32, 12 … Σ 1, 2, 10, 44, 214, 1052, 5284, 26840, …
  (−∞, −1]nonnegative
• A028326 → Veronese 2 → θ+1
  2 · 2 · 2, 4 · 2, 12 … Σ 2, 2, 6, 14, 32, 72, 160, 352, …
  (−∞, 0]nonnegative
• A172431 → basis A007318
  1 · 3, 2 · 8, 10, 3 · 21, 38, 22, 4 … Σ 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729 → basis A007318 → D+1
  1 · 2, 1 · 3, 4, 1 · 4, 9, 6, 1 … Σ 1, 3, 8, 20, 48, 112, 256, 576, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … Σ 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1]rational roots
• A111125 → basis A013609 → reverse
  1 · 2, 4 · 4, 14, 11 · 8, 40, 62, 29 … Σ 1, 6, 29, 139, 666, 3191, 15289, 73254, …
  (−∞, 0]nonnegative
• A013620 → D
  3 · 12, 18 · 36, 108, 81 · 96, 432, 648, 324 … Σ 0, 3, 30, 225, 1500, 9375, 56250, …
  (−∞, 0]nonnegativerational roots
• A272867 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 6, 18, 13 · 36, 216, 480, 468, 169 … Σ 1, 37, 1369, 50653, 1874161, 69343957, …
  [−1, 0]nonnegative
• A145324 → basis A013610 → reverse
  1 · 6, 3 · 54, 51, 12 · 648, 882, 399, 60 … Σ 1, 9, 117, 1989, 41769, 1044225, …
  (−∞, 0]nonnegativerational roots
• A013620 → basis A013611 → θ+1
  1 · 5, 24 · 25, 240, 432 · 125, 1800, 6480, 6912 … Σ 1, 29, 697, 15317, 319345, 6431117, …
  (−∞, 0]nonnegativerational roots
• A207823 → reverse → x ↦ x+1
  1 · 5, 4 · 24, 38, 15 · 115, 272, 214, 56 … Σ 1, 9, 77, 657, 5605, 47817, 407933, …
  (−∞, −1]nonnegative
• A154233: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 7, 1 · 1, 19, 19, 1 … Σ 1, 2, 9, 40, 251, 1662, 13615, 118640, …
  (−∞, 0)nonnegative
• A308737 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 10, 16 · 1, 20, 68, 64 … Σ 1, 5, 27, 153, 891, 5265, 31347, …
  [−1, 0]nonnegativerational roots
• A094305 → θ+1
  1 · 3, 6 · 6, 24, 18 · 10, 60, 90, 40 … Σ 1, 9, 48, 200, 720, 2352, 7168, 20736, …
  (−∞, 0]nonnegativerational roots
• A063007 → basis A013609 → D+1
  1 · 7, 4 · 49, 84, 24 · 327, 984, 840, 160 … Σ 1, 11, 157, 2311, 34121, 502387, …
  (−∞, 0]nonnegative
• A128813 → basis A013609 → θ+1
  1 · 2, 4 · 8, 40, 36 · 56, 472, 972, 576 … Σ 1, 6, 84, 2076, 79556, 4366376, …
  (−∞, 0]nonnegative
• A143495 → reverse → x ↦ x/(x+1)
  1 · 1, 4 · 1, 9, 17 · 1, 15, 64, 77 … Σ 1, 5, 27, 157, 979, 6517, 46107, …
  [−1, 0]nonnegative
• A038259 → basis A013611 → θ+1
  1 · 11, 40 · 121, 880, 1200 … Σ 1, 51, 2201, 87451, 3306801, 120981251, …
  (−∞, 0]nonnegativerational roots
• A347823 → basis A007318
  1 · 5, 3 · 16, 18, 5 · 44, 72, 39, 7 … Σ 1, 8, 39, 162, 621, 2268, 8019, 27702, …
  (−∞, −1]nonnegativerational roots
• A359363 → x ↦ x+1
  1 · 1, 1 · 2, 3, 1 · 6, 12, 7, 1 … Σ 1, 2, 6, 26, 138, 834, 5526, 39218, …
  (−∞, −1]nonnegative
• A372873 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 1, 5 · 0, 0, 6, 14 … Σ 1, 2, 6, 20, 68, 232, 792, 2704, …
  [−1, 0]nonnegative