Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• type_b_eulerian → γ (scale 1/4) → x ↦ x+1 → θ+1
  1 · 1 · 2, 2 · 6, 10 … Σ 1, 1, 4, 16, 95, 663, 5590, 54218, …
  (−∞, 0]nonnegative
• A122848 → basis A013610 → θ+1
  1 · 1, 6 · 2, 18, 27 · 4, 54, 162, 108 … Σ 1, 7, 47, 328, 2392, 18224, 144688, …
  (−∞, 0]nonnegative
• A091042 → basis A007318 → θ+1
  1 · 4, 6 · 16, 40, 15 · 64, 224, 168, 28 … Σ 1, 10, 71, 484, 3229, 21198, 137411, …
  (−∞, 0]nonnegative
• A271703 → D
  1 · 2, 2 · 6, 12, 3 · 24, 72, 36, 4 … Σ 0, 1, 4, 21, 136, 1045, 9276, 93289, …
  (−∞, 0]nonnegative
• A064861 → D+1
  4, 7, 2 · 6, 21, 20, 4 · 7, 32, 49, 28, 4 … Σ 13, 51, 120, 432, 972, 3348, 7344, …
  (−∞, 0]nonnegative
• A109692 → basis A007318 → (θ+1)²
  1 · 2, 4 · 8, 40, 27 · 48, 400, 612, 240 … Σ 1, 6, 75, 1300, 28285, 735946, …
  (−∞, 0]nonnegative
• A110552 → (θ+1)²
  1 · 1, 8 · 1, 20, 36 · 1, 40, 153, 128 … Σ 1, 9, 57, 322, 1720, 8864, 44464, …
  (−∞, 0]nonnegative
• A207607 → basis A013611 → D+1
  1 · 11, 8 · 44, 100, 32 · 153, 612, 624, 128 … Σ 1, 19, 176, 1517, 12527, 100456, …
  (−∞, 0]nonnegative
• basis-map coefficients of A127647 → (θ+1)²
  1 · 1, 4 · 2, 16, 18 · 3, 36, 81, 48 … Σ 1, 5, 36, 168, 800, 3456, 14560, 59136, …
  (−∞, 0]nonnegative
• A038320 → basis A013610 → x ↦ x+1
  1225, 1260, 324 · 42875, 66150, 34020, 5832 … Σ 2809, 148877, 7890481, 418195493, …
  (−∞, −1]nonnegativerational roots
• A169803 → D → (θ+1)²
  1 · 2 · 3, 8 · 4, 24 … Σ 0, 1, 2, 11, 28, 80, 194, 461, …
  (−∞, 0]nonnegative
• A034870 → Veronese 2 → θ+1
  1 · 1, 2 · 1, 12, 3 · 1, 30, 45, 4 … Σ 1, 3, 16, 80, 384, 1792, 8192, 36864, …
  (−∞, 0]nonnegative
• narayana → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 18, 4 … Σ 1, 3, 10, 35, 126, 462, 1716, 6435, …
  (−∞, 0]nonnegative
• A154986 → γ → θ+1
  1 · 1 · 1, 4 · 1, 16 … Σ 1, 1, 5, 17, 113, 633, 5373, 41409, …
  (−∞, 0]nonnegative
• A142460 → (θ+1)²
  1 · 1, 4 · 1, 48, 9 · 1, 332, 747, 16 … Σ 1, 5, 58, 1096, 26740, 797832, …
  (−∞, 0]nonnegative
• A225471 → basis A013611 → D
  4 · 48, 32 · 704, 768, 192 · 12800, 17920, 7680, 1024 … Σ 0, 4, 80, 1664, 39424, 1069056, …
  (−∞, 0]nonnegative
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1
  1 · 4, 2 · 14, 14, 3 · 44, 66, 30, 4 … Σ 1, 6, 31, 144, 621, 2538, 9963, 37908, …
  (−∞, −1]nonnegative
• A008459 → D
  1 · 4, 2 · 9, 18, 3 · 16, 72, 48, 4 … Σ 0, 1, 6, 30, 140, 630, 2772, 12012, …
  (−∞, 0]nonnegative
• A257611 → D+1
  1 · 6, 3 · 39, 48, 9 · 240, 639, 294, 27 … Σ 1, 9, 96, 1200, 17280, 282240, 5160960, …
  (−∞, 0]nonnegative
• A008517 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 10, 15 · 1, 25, 105, 105 … Σ 1, 4, 26, 236, 2752, 39208, 660032, …
  [−1, 0]nonnegative
• A086645 → ⊙ C(n+1,k) (shift +2)
  1, 12, 1 · 1, 45, 45, 1 · 1, 112, 420, 112, 1 … Σ 14, 92, 646, 4652, 34124, 253528, …
  (−∞, 0]nonnegative
• A008277 → D → x ↦ x+1
  1 · 5, 2 · 22, 18, 3 · 99, 122, 42, 4 … Σ 0, 1, 7, 43, 267, 1731, 11819, 85107, …
  (−∞, −1]nonnegative
• A286718 → basis A013610 → D+1
  1 · 5, 3 · 31, 39, 9 · 278, 468, 216, 27 … Σ 1, 8, 79, 989, 15166, 276302, 5840200, …
  (−∞, 0]nonnegative
• A145324 → θ+1
  1 · 1, 4 · 1, 10, 18 · 1, 18, 78, 96 … Σ 1, 5, 29, 193, 1458, 12366, 116568, …
  (−∞, 0]nonnegative
• A359363 → D+1
  1 · 1, 1 · 1, 3, 1 · 1, 9, 7, 1 … Σ 1, 2, 5, 18, 77, 368, 1899, 10370, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/2) (shift +1) → D
  1 · 4 · 11, 8 · 26, 68 … Σ 0, 0, 1, 4, 19, 94, 519, 3144, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → D+1
  1 · 1, 1 · 0, 2, 1 · 0, 0, 3, 1 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0]nonnegativerational roots
• A166960 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 8, 8 · 1, 30, 78, 50 … Σ 1, 3, 17, 159, 2109, 37539, 859065, …
  [−1, 0]nonnegative
• A008288 → basis A013609 → D
  2 · 10, 8 · 36, 64, 24 · 116, 320, 264, 64 … Σ 0, 2, 18, 124, 764, 4422, 24598, …
  (−∞, 0]nonnegative
• A038333 → x ↦ x+1 → θ+1
  361, 532, 147 · 6859, 15162, 8379, 1372 … Σ 1040, 31772, 949104, 27875536, …
  (−∞, 0]nonnegativerational roots
• A201637 → reverse (shift +1) → D+1
  1 · 1, 1 · 2, 4, 1 · 6, 22, 11, 1 … Σ 1, 2, 7, 40, 315, 3150, 38115, 540540, …
  (−∞, 0]nonnegative
• A145324 → basis A013609 → D
  4 · 34, 48 · 266, 784, 576 · 2196, 10024, 15216, 7680 … Σ 0, 4, 82, 1626, 35116, 841924, …
  (−∞, 0]nonnegative
• A038330 → basis A013611
  256, 512, 256 · 4096, 12288, 12288, 4096 … Σ 1024, 32768, 1048576, 33554432, …
  (−∞, 0]nonnegativerational roots
• A003506 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 12, 12 · 4, 24, 48, 32 … Σ 1, 6, 27, 108, 405, 1458, 5103, 17496, …
  [−1, 0]nonnegativerational roots
• A330510 → (θ+1)²
  2 · 3, 12 · 4, 40, 36 · 5, 88, 198, 80 … Σ 2, 15, 80, 371, 1586, 6419, 24968, …
  (−∞, 0]nonnegative
• A054142 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 18, 4 … Σ 1, 3, 10, 33, 105, 324, 977, 2895, …
  (−∞, 0]nonnegative
• A049374 → x ↦ x/(x+1)
  1 · 6, 7 · 42, 102, 61 · 336, 1284, 1596, 649 … Σ 1, 13, 205, 3865, 85945, 2221285, …
  [−1, 0]nonnegative
• A285061 → D
  4 · 24, 32 · 124, 480, 192 · 624, 5312, 5376, 1024 … Σ 0, 4, 56, 796, 12336, 208884, 3844072, …
  (−∞, 0]nonnegative
• A200536 → basis A007318 → reverse
  1 · 2, 7, 6 · 4, 28, 73, 84, 36 … Σ 1, 15, 225, 3375, 50625, 759375, …
  (−∞, 0]nonnegativerational roots
• A125092 → reverse → (θ+1)²
  1 · 4, 4 · 9, 32, 9 · 16, 108, 108, 16 … Σ 1, 8, 50, 248, 1048, 3952, 13696, …
  (−∞, 0]nonnegative
• A269947 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 18, 29, 12, 1 … Σ 1, 2, 6, 60, 1740, 114840, 14584680, …
  (−∞, −1]nonnegativerational roots
• A207607 → basis A013609 → θ+1
  1 · 3, 8 · 8, 36, 24 · 21, 132, 180, 64 … Σ 1, 11, 68, 397, 2243, 12380, 67141, …
  (−∞, 0]nonnegative
• A290319 → basis A013610 → D
  3 · 24, 18 · 276, 324, 81 · 4224, 6192, 2592, 324 … Σ 0, 3, 42, 681, 13332, 309807, 8371710, …
  (−∞, 0]nonnegative
• A066325: Hermite he
  1 · 0, 1 · -1, 0, 1 · 0, -3, 0, 1 … Σ 1, 1, 0, -2, -2, 6, 16, -20, …
  (−∞, ∞)
• A038326 → reverse
  1 · 12, 11 · 144, 264, 121 · 1728, 4752, 4356, 1331 … Σ 1, 23, 529, 12167, 279841, 6436343, …
  (−∞, 0]nonnegativerational roots
• A144440 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 13, 13 · 1, 57, 165, 110 … Σ 1, 3, 27, 333, 5463, 112509, 2794959, …
  [−1, 0]nonnegative
• A128813 → D
  1 · 4, 6 · 10, 54, 54 · 20, 254, 864, 720 … Σ 0, 1, 10, 118, 1858, 38968, 1064272, …
  (−∞, 0]nonnegative
• A099097 → x ↦ x−1
  1 · -3, 3 · 8, -17, 9 · -21, 69, -75, 27 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (5/9, 1]
• A035469 → reverse
  1 · 1 · 1, 4 · 1, 12, 28 … Σ 1, 1, 5, 41, 465, 6721, 117941, …
  (−∞, 0]nonnegative
• A358735 → D+1
  1 · 2, 1 · 5, 8, 2 · 11, 42, 34, 6 … Σ 1, 3, 15, 93, 675, 5587, 51845, 532705, …
  (−∞, 0]nonnegative