Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A173018 → reverse (shift +1) → D+1
  1 · 1, 1 · 1, 3, 1 · 1, 9, 7, 1 … Σ 1, 2, 5, 18, 84, 480, 3240, 25200, …
  (−∞, 0]nonnegative
• A038325 → basis A007318 → (θ+1)²
  1 · 22, 44 · 484, 1936, 1089 … Σ 1, 66, 3509, 167706, 7510833, …
  (−∞, 0]nonnegative
• A078812 → x ↦ x+1 → D
  1 · 6, 2 · 25, 18, 3 · 90, 102, 36, 4 … Σ 0, 1, 8, 46, 232, 1091, 4912, 21468, …
  (−∞, −1]nonnegative
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → x ↦ x/(x+1)
  1 · 4, 6 · 14, 42, 31 · 44, 198, 294, 144 … Σ 1, 10, 87, 680, 4925, 33750, 221875, …
  [−1, 0]nonnegative
• basis-map coefficients of A030528, shifted → θ+1 → reverse
  1 · -2 · 3 · -4 … Σ 1, -2, 3, -4, 5, -6, 7, -8, …
  (−∞, 0]rational roots
• A091042 → basis A013609 → x ↦ x/(x+1)
  1 · 4, 10 · 16, 72, 76 · 64, 416, 864, 568 … Σ 1, 14, 164, 1912, 22288, 259808, …
  [−1, 0]nonnegative
• A038327 → (θ+1)²
  1 · 12, 4 · 144, 96, 9 · 1728, 1728, 324, 16 … Σ 1, 16, 249, 3796, 56953, 843648, …
  (−∞, 0]nonnegative
• A038220 → Veronese 2 → D
  4 · 36 · 216, 32 · 1080, 480 … Σ 0, 0, 4, 36, 248, 1560, 9372, 54684, …
  (−∞, 0]nonnegative
• A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
  1 · 1, 1 · 1, 4, 2 · 1, 9, 18, 6 … Σ 1, 2, 7, 34, 209, 1546, 13327, 130922, …
  (−∞, 0)nonnegative
• A159854 → basis A013609 → reverse
  4, 4, 1 · 8, 16, 10, 2 · 16, 48, 52, 24, 4 … Σ 9, 36, 144, 576, 2304, 9216, 36864, …
  (−∞, 0]nonnegativerational roots
• A161198 → basis A008277 → x ↦ x+1
  1 · 5, 2 · 39, 28, 4 · 407, 402, 108, 8 … Σ 1, 7, 71, 925, 14593, 269039, 5658823, …
  (−∞, −1]nonnegative
• A132393 → basis A013609 → (θ+1)²
  1 · 1, 8 · 2, 24, 36 · 6, 88, 216, 128 … Σ 1, 9, 62, 438, 3364, 28444, 264704, …
  (−∞, 0]nonnegative
• A013610 → basis A013611 → x ↦ x/(x+1)
  1 · 4, 16 · 16, 128, 256 · 64, 768, 3072, 4096 … Σ 1, 20, 400, 8000, 160000, 3200000, …
  [−1, 0]nonnegativerational roots
• A130123 → basis A008277 → (θ+1)²
  1 · 2, 8 · 4, 48, 36 · 8, 224, 432, 128 … Σ 1, 10, 88, 792, 7536, 76352, 823488, …
  (−∞, 0]nonnegative
• A089231 → x ↦ x+1 → θ+1
  1 · 3, 4 · 13, 36, 18 · 73, 312, 324, 96 … Σ 1, 7, 67, 805, 11581, 193531, 3677647, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → D → θ+1
  -1 · 0, 4 · 0, 0, -9 · 0, 0, 0, 16 … Σ 0, -1, 4, -9, 16, -25, 36, -49, …
  (−∞, 0]rational roots
• A085880 → basis A013610 → (θ+1)²
  1 · 2, 12 · 8, 96, 162 · 40, 720, 2430, 2160 … Σ 1, 14, 266, 5350, 109550, 2257500, …
  (−∞, 0]nonnegative
• A143495 → D+1 → (θ+1)²
  1 · 4, 4 · 16, 36, 9 · 64, 244, 135, 16 … Σ 1, 8, 61, 459, 3468, 26569, 207609, …
  (−∞, 0]nonnegative
• A026729 → basis A008277 → D
  1 · 4, 2 · 13, 16, 3 · 39, 88, 39, 4 … Σ 0, 1, 6, 32, 170, 933, 5357, 32289, …
  (−∞, 0]nonnegative
• A093375 → D → θ+1
  1 · 4, 4 · 9, 24, 9 · 16, 72, 72, 16 … Σ 0, 1, 8, 42, 176, 640, 2112, 6496, …
  (−∞, 0]nonnegative
• A109692 → D
  1 · 4, 6 · 9, 46, 45 · 16, 172, 528, 420 … Σ 0, 1, 10, 100, 1136, 14816, 220032, …
  (−∞, 0]nonnegative
• A121574 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 4 · 10, 27, 18 · 33, 134, 180, 80 … Σ 1, 7, 55, 427, 3319, 25795, 200479, …
  [−1, 0]nonnegative
• A124932 → basis A007318 → D
  3 · 21, 12 · 96, 108, 30 · 360, 600, 330, 60 … Σ 0, 3, 33, 234, 1350, 6885, 32319, …
  (−∞, 0]nonnegativerational roots
• A110441 → basis A007318 → basis A007318
  1 · 5, 1 · 23, 10, 1 · 105, 71, 15, 1 … Σ 1, 6, 34, 192, 1084, 6120, 34552, …
  (−8, −2)nonnegative
• A163626 → D+1
  1 · 0, -1 · -2, 1, 2 · -6, 17, -6, -6 … Σ 1, -1, 1, -1, 1, -1, 1, -1, …
  (−∞, ∞)
• A269939 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 4 · 0, 1, 12, 26 … Σ 1, 1, 5, 39, 423, 5889, 100125, …
  [−1, 0]nonnegative
• A133087 → Veronese 2 → x ↦ x+1
  1 · 2 · 2, 1 · 6, 4 … Σ 1, 2, 3, 10, 17, 58, 99, 338, …
  (−∞, −1]nonnegative
• A391045 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 12, 31, 21 … Σ 1, 3, 13, 65, 353, 2023, 12041, 73713, …
  [−1, 0]nonnegative
• A139548 → basis A013610 → x ↦ x/(x+1)
  1 · 8, 40, 50 · 64, 640, 2400, 4000, 2500 … Σ 1, 98, 9604, 941192, 92236816, …
  [−1, 0]nonnegativerational roots
• A063967 → basis A013609 → (θ+1)²
  1 · 2, 8 · 6, 40, 36 · 16, 160, 288, 128 … Σ 1, 10, 82, 592, 3976, 25408, 156608, …
  (−∞, 0]nonnegative
• A249247 → D
  1 · 3 · 10, 2 · 42, 14 … Σ 0, 1, 3, 12, 56, 309, 1983, 14572, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → basis A013610
  1 · -2, -3 · 4, 12, 9 · -8, -36, -54, -27 … Σ 1, -5, 25, -125, 625, -3125, 15625, …
  [−1, 0]rational roots
• A038325 → basis A013610
  1 · 22, 33 · 484, 1452, 1089 … Σ 1, 55, 3025, 166375, 9150625, …
  [−1, 0]nonnegativerational roots
• A356546 → basis A013611 → (θ+1)²
  1 · 4, 32 · 24, 384, 864 · 160, 3840, 17280, 20480 … Σ 1, 36, 1272, 41760, 1300320, 38973312, …
  (−∞, 0]nonnegative
• A178821 → basis A013611 → (θ+1)²
  1 · 10, 80 · 60, 960, 2160 · 280, 6720, 30240, 35840 … Σ 1, 90, 3180, 73080, 1300320, 19486656, …
  (−∞, 0]nonnegative
• A094638 → (θ+1)²
  1 · 1, 4 · 1, 12, 18 · 1, 24, 99, 96 … Σ 1, 5, 31, 220, 1756, 15596, 152804, …
  (−∞, 0]nonnegative
• A038271 → basis A007318 → x ↦ x/(x+1)
  1 · 12, 17 · 144, 408, 289 · 1728, 7344, 10404, 4913 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  [−1, 0]nonnegativerational roots
• A007318 → −D²+(−1+n)D+1+n → θ+1
  1 · 2, 4 · 3, 16, 9 · 4, 36, 54, 16 … Σ 1, 6, 28, 110, 384, 1232, 3712, 10656, …
  (−∞, 0]nonnegative
• A038284 → basis A013611 → D+1
  868, 1824, 576 · 16856, 62496, 65664, 13824 … Σ 3268, 158840, 7352848, 329451488, …
  (−∞, 0]nonnegativerational roots
• A130534 → Veronese 2 → x ↦ x+1
  1 · 1 · 3, 1 · 12, 6 … Σ 1, 1, 4, 18, 98, 630, 4676, 39368, …
  (−∞, −1]nonnegative
• A038334 → x ↦ x/(x+1)
  144, 480, 400 · 1728, 8640, 14400, 8000 … Σ 1024, 32768, 1048576, 33554432, …
  [−1, 0]nonnegativerational roots
• A085880 → D → D
  4 · 30, 30 · 168, 336, 168 · 840, 2520, 2520, 840 … Σ 0, 0, 4, 60, 672, 6720, 63360, 576576, …
  (−∞, 0]nonnegativerational roots
• A008277 → θ+2 (shift +1) → θ+1
  2 · 2, 6 · 2, 18, 12 · 2, 42, 72, 20 … Σ 2, 8, 32, 136, 622, 3060, 16130, 90688, …
  (−∞, 0]nonnegative
• A038329 → basis A013610 → θ+1
  225, 540, 243 · 3375, 12150, 10935, 2916 … Σ 1008, 29376, 829440, 22892544, …
  (−∞, 0]nonnegativerational roots
• row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between…
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0)nonnegative
• A356546 → basis A013610
  1 · 4, 6 · 24, 72, 54 · 160, 720, 1080, 540 … Σ 1, 10, 150, 2500, 43750, 787500, …
  [−1, 0]nonnegativerational roots
• A201701: Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 17, 41, 99, 239, …
  (−∞, −1]nonnegative
• A145596 → x ↦ x+1 → (θ+1)²
  1 · 4, 8 · 14, 56, 27 · 48, 288, 288, 64 … Σ 1, 12, 97, 688, 4595, 29688, 187915, …
  (−∞, 0]nonnegative
• A028338 → basis A013610 → reverse
  1 · 3, 2 · 9, 18, 8 · 27, 108, 132, 48 … Σ 1, 5, 35, 315, 3465, 45045, 675675, …
  (−∞, 0]nonnegativerational roots
• A063007 → x ↦ x+1 → θ+1
  1 · 3, 4 · 13, 36, 18 · 63, 264, 270, 80 … Σ 1, 7, 67, 677, 6941, 71435, 735295, …
  (−∞, 0]nonnegative