Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038301 → basis A013611 → D
  1760, 3872 · 52800, 232320, 255552 … Σ 5632, 540672, 46137344, 3690987520, …
  (−∞, 0]nonnegativerational roots
• A143491 → basis A013609 → basis A007318
  30, 22, 4 · 210, 214, 72, 8 … Σ 56, 504, 5040, 55440, 665280, 8648640, …
  (−∞, −1]nonnegativerational roots
• A272866 → Veronese 2
  1 · 1, 1 · 1, 11, 1 · 1, 30, 30, 1 … Σ 1, 2, 13, 62, 313, 1562, 7813, 39062, …
  (−∞, 0]nonnegative
• A269947 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 32, 81, 16 … Σ 1, 4, 13, 129, 3724, 245232, 31106420, …
  (−∞, 0]nonnegative
• A084950 → D+1
  1 · 3, 1 · 10, 4 · 42, 20, 1 … Σ 1, 4, 14, 63, 339, 2140, 15544, 127885, …
  (−∞, 0]nonnegative
• A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder p…
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … Σ 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, 0)nonnegative
• A114655 → θ+1 → reverse
  2 · 4, 4 · 6, 24, 8 · 8, 72, 96, 16 … Σ 2, 8, 38, 192, 1002, 5336, 28814, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013611 → x ↦ x/(x+1)
  1 · 1, 10, 25 · 1, 20, 150, 500, 625 … Σ 1, 36, 1296, 46656, 1679616, 60466176, …
  [−1, 0]nonnegativerational roots
• A166979 → reverse
  1, 11, 1 · 1, 30, 111, 1 · 1, 58, 681, 1111, 1 … Σ 13, 143, 1852, 27563, 473725, 9290396, …
  (−∞, 0]nonnegative
• A113214 → θ+1 → (θ+1)²
  1 · 2, 8 · 0, 24, 27 · 0, 16, 108, 64 … Σ 1, 10, 51, 188, 580, 1596, 4053, 9704, …
  (−∞, 0]nonnegative
• A152198 → D+1
  1 · 1 · 2, 1 · 2, 1 … Σ 1, 1, 3, 3, 8, 8, 20, 20, …
  (−∞, 0]nonnegativerational roots
• A257616 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 40, 40 · 8, 336, 960, 640 … Σ 1, 6, 84, 1944, 61776, 2496096, …
  [−1, 0]nonnegative
• A136394 → D+1
  1 · 1 · 2, 1 · 6, 5 … Σ 1, 1, 3, 11, 50, 274, 1764, 13068, …
  (−∞, 0]nonnegative
• A013620 → basis A008288 → D+1
  1 · 8, 3 · 64, 57, 9 · 458, 711, 270, 27 … Σ 1, 11, 130, 1466, 15991, 170357, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → D+1
  1 · 2 · 6, 2 · 24, 16 … Σ 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0]nonnegative
• A038325 → basis A007318 → θ+1
  1 · 22, 22 · 484, 968, 363 … Σ 1, 44, 1815, 71874, 2767149, 104361048, …
  (−∞, 0]nonnegativerational roots
• A126351 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 12, 40, 37 … Σ 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  [−1, 0]nonnegative
• A128813 → basis A013609 → D+1
  1 · 4, 2 · 28, 44, 12 · 292, 884, 756, 144 … Σ 1, 6, 84, 2076, 79556, 4366376, …
  (−∞, 0]nonnegative
• A123519 → reverse → θ+1
  4, 12, 3 · 8, 40, 36, 4 · 16, 112, 180, 80, 5 … Σ 19, 88, 393, 1708, 7275, 30512, 126417, …
  (−∞, 0]nonnegative
• A113953 → x ↦ x−2
  1 · -2, 1 · 0, -2, 1 · 8, -4, -2, 1 … Σ 1, -1, -1, 3, -1, -5, 7, 3, …
  (−6, 2]
• basis-map coefficients of A030528, shifted → θ+1 → D
  -2 · 0, 6 · 0, 0, -12 · 0, 0, 0, 20 … Σ 0, -2, 6, -12, 20, -30, 42, -56, …
  (−∞, 0]rational roots
• A119900 → θ+1
  1 · 0, 4 · 0, 2, 9 · 0, 0, 12, 16 … Σ 1, 4, 11, 28, 68, 160, 368, 832, …
  (−∞, 0]nonnegative
• A120909 → basis A013610 → D+1
  3 · 27, 18 · 135, 324, 108 · 567, 2430, 2916, 648 … Σ 3, 45, 567, 6561, 72171, 767637, …
  (−∞, 0]nonnegativerational roots
• A290319 → basis A013609 → θ+1
  1 · 2, 4 · 12, 32, 12 · 120, 368, 216, 32 … Σ 1, 6, 56, 736, 12544, 263168, 6561792, …
  (−∞, 0]nonnegative
• A094305 → x ↦ x+1 → reverse
  1 · 3, 6 · 6, 24, 24 · 10, 60, 120, 80 … Σ 1, 9, 54, 270, 1215, 5103, 20412, …
  (−∞, 0]nonnegativerational roots
• A133081 → reverse (shift +1) → (θ+1)²
  1 · 0, 4, 9 · 0, 0, 0, 0, 25 · 0, 0, 0, 0, 0, 36, 49 … Σ 1, 13, 25, 85, 81, 221, 169, 421, …
  (−∞, 0]nonnegativerational roots
• A059364 → basis A013609 → D
  2 · 22, 16 · 244, 376, 144 · 3036, 7288, 5808, 1536 … Σ 0, 2, 38, 764, 17668, 470478, 14268090, …
  (−∞, 0]nonnegative
• A027467 → basis A013609 → θ+1
  1 · 16, 4 · 256, 128, 12 · 4096, 3072, 576, 32 … Σ 1, 20, 396, 7776, 151632, 2939328, …
  (−∞, 0]nonnegativerational roots
• A104684 → basis A007318 → reverse
  1 · 1, 3 · 1, 8, 13 · 1, 15, 57, 63 … Σ 1, 4, 22, 136, 886, 5944, 40636, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 15, 23, 10, 1 … Σ 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, −1]nonnegative
• A001263 → D → (θ+1)²
  1 · 3, 8 · 6, 48, 27 · 10, 160, 270, 64 … Σ 0, 1, 11, 81, 504, 2850, 15147, 77077, …
  (−∞, 0]nonnegative
• A123519 → basis A007318 → basis A007318
  29, 22, 4 · 169, 188, 68, 8 · 985, 1444, 780, 184, 16 … Σ 55, 433, 3409, 26839, 211303, 1663585, …
  (−4, −2)nonnegative
• A011973 → D+1
  1 · 1 · 2, 1 · 3, 2 … Σ 1, 1, 3, 5, 10, 18, 33, 59, …
  (−∞, 0]nonnegative
• A305540 → D+1
  1 · 2, 1 · 3, 2 · 5, 10, 3 … Σ 1, 3, 5, 18, 31, 132, 233, 1152, …
  (−∞, 0]nonnegative
• A075504 → D
  27, 2 · 567, 108, 3 · 10935, 4050, 270, 4 … Σ 29, 678, 15259, 350951, 8481570, …
  (−∞, 0]nonnegative
• A272866 → Veronese 2 → D+1
  1 · 2, 1 · 12, 13, 1 · 31, 90, 33, 1 … Σ 1, 3, 26, 155, 939, 5467, 31252, …
  (−∞, 0]nonnegative
• A027467 → basis A013609 → D
  2 · 64, 8 · 1536, 384, 24 · 32768, 12288, 1536, 64 … Σ 0, 2, 72, 1944, 46656, 1049760, …
  (−∞, 0]nonnegativerational roots
• A225471 → basis A013610 → (θ+1)²
  1 · 4, 12 · 32, 144, 81 · 384, 2112, 1944, 432 … Σ 1, 16, 257, 4872, 109209, 2848776, …
  (−∞, 0]nonnegative
• Hadamard-factor parent of A155495 → D+1
  7, 8, 1 · 16, 45, 18, 1 · 29, 168, 154, 32, 1 … Σ 16, 80, 384, 1792, 8192, 36864, 163840, …
  (−∞, 0]nonnegative
• A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
  1 · 5, 5 · 15, 30, 15 · 35, 105, 105, 35 … Σ 1, 10, 60, 280, 1120, 4032, 13440, …
  (−∞, 0)nonnegativerational roots
• A086645 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 8, 48, 56 · 32, 288, 768, 576 … Σ 1, 8, 112, 1664, 24832, 370688, …
  [−1, 0]nonnegative
• A008292 → θ+1 (shift +2) → x ↦ x+1
  12, 14, 3 · 60, 100, 45, 4 · 360, 780, 540, 124, 5 … Σ 29, 209, 1809, 18269, 210853, 2737769, …
  (−∞, −1]nonnegative
• A038220 → Veronese 2 (j=1) → D+1
  2 · 12 · 62, 8 · 312, 96 … Σ 0, 2, 12, 70, 408, 2346, 13284, 74222, …
  (−∞, 0]nonnegative
• A357367 → x ↦ x+1
  1 · 2, 2 · 18, 30, 12 · 264, 624, 480, 120 … Σ 1, 4, 60, 1488, 51600, 2299680, …
  (−∞, −1]nonnegative
• A034870 → Veronese 2 → basis A007318
  1 · 2, 1 · 8, 8, 1 · 32, 48, 18, 1 … Σ 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, −1]nonnegative
• A169803 → basis A013610 → D
  3 · 6 · 15, 18 · 30, 54 … Σ 0, 3, 6, 33, 84, 303, 834, 2589, …
  (−∞, 0]nonnegative
• A133214 → basis A013609 → x ↦ x/(x+1)
  13, 58, 61 · 63, 417, 885, 595 … Σ 132, 1960, 30736, 496800, 8185920, …
  [−1, 0]nonnegative
• A034870 → basis A013609 → (θ+1)²
  1 · 4, 32, 36 · 16, 256, 864, 1024, 400 … Σ 1, 72, 2560, 71680, 1769472, 40370176, …
  (−∞, 0]nonnegative
• A172431: Even row Pascal-square read by antidiagonals.
  1 · 1, 2 · 1, 4, 3 · 1, 6, 10, 4 … Σ 1, 3, 8, 21, 55, 144, 377, 987, …
  (−∞, 0)nonnegative
• A091044 → basis A013610 → θ+1
  1 · 4, 12 · 16, 96, 81 · 64, 576, 1080, 432 … Σ 1, 16, 193, 2152, 23065, 240808, …
  (−∞, 0]nonnegative