Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A178821 → basis A013610 → (θ+1)²
  1 · 10, 60 · 60, 720, 1215 · 280, 5040, 17010, 15120 … Σ 1, 70, 1995, 37450, 547750, 6772500, …
  (−∞, 0]nonnegative
• A103633 → Veronese 2 → θ+1
  1 · 0, 2 · 0, 2 · 0, 2, 3 … Σ 1, 0, 2, 2, 5, 6, 13, 15, …
  (−∞, 0]nonnegative
• A113953 → basis A013609 → reverse
  1 · 2, 1 · 4, 8, 3 · 8, 28, 22, 5 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, 0]nonnegative
• A185410: A decomposition of the double factorials A001147.
  1 · 1 · 1, 2 · 1, 10, 4 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0)nonnegative
• basis-map coefficients of A371080, shifted → basis A013609 → D
  2 · 10, 8 · 78, 96, 24 · 836, 1272, 528, 64 … Σ 0, 2, 18, 198, 2700, 44388, 857304, …
  (−∞, 0]nonnegative
• A062264 → reverse → x ↦ x+1
  1 · 6, 1 · 28, 14, 1 · 120, 108, 24, 1 … Σ 1, 7, 43, 253, 1462, 8378, 47818, …
  (−∞, −1]nonnegative
• A105278 → x ↦ x+1
  1 · 3, 1 · 13, 8, 1 · 73, 63, 15, 1 … Σ 1, 4, 22, 152, 1256, 12032, 130768, …
  (−∞, −1]nonnegative
• A038277 → basis A013609 → x ↦ x/(x+1)
  1 · 18, 40 · 324, 1440, 1600 … Σ 1, 58, 3364, 195112, 11316496, …
  [−1, 0]nonnegativerational roots
• A305540 → (θ+1)²
  1 · 1, 4 · 1, 8 · 1, 16, 27 … Σ 1, 5, 9, 44, 79, 422, 765, 4538, …
  (−∞, 0]nonnegative
• A196347 → x ↦ x+1 → basis A007318
  1 · 3, 1 · 18, 12, 2 · 162, 162, 54, 6 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
  (−∞, −1]nonnegativerational roots
• A133081 → θ+1
  1 · 1 · 0, 2, 3 · 0, 0, 3 … Σ 1, 1, 5, 3, 9, 5, 13, 7, …
  (−∞, 0]nonnegativerational roots
• A142075 → D+1
  1 · 3, 2 · 9, 16, 4 · 23, 110, 68, 8 … Σ 1, 5, 29, 209, 1809, 18269, 210853, …
  (−∞, 0]nonnegative
• A232535 → reverse
  1 · 2, 1 · 3, 8, 1 · 4, 25, 18, 1 … Σ 1, 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, 0]nonnegative
• A161198 → basis A013611 → D
  8 · 64, 128 · 568, 1920, 1536 … Σ 0, 8, 192, 4024, 85568, 1917512, …
  (−∞, 0]nonnegative
• A038333 → x ↦ x+1
  361, 266, 49 · 6859, 7581, 2793, 343 … Σ 676, 17576, 456976, 11881376, …
  (−∞, −1]nonnegativerational roots
• A038226 → basis A013610 → basis A007318
  1225, 1680, 576 · 42875, 88200, 60480, 13824 … Σ 3481, 205379, 12117361, 714924299, …
  (−∞, −1]nonnegativerational roots
• A130534 → Veronese 2 (j=1) → x ↦ x+1
  1 · 3 · 12, 1 · 60, 10 … Σ 0, 1, 3, 13, 70, 448, 3318, 27902, …
  (−∞, −1]nonnegative
• A124038 → D
  1 · -2, 2 · -2, -4, 3 · 4, -6, -6, 4 … Σ 0, 1, 0, -3, -4, 0, 6, 7, …
  (−∞, ∞)
• A109692 → (θ+1)²
  1 · 1, 4 · 1, 16, 27 · 1, 36, 207, 240 … Σ 1, 5, 44, 484, 6280, 93616, 1576064, …
  (−∞, 0]nonnegative
• A038322 → basis A013609 → x ↦ x+1
  1 · 35, 16 · 1225, 1120, 256 … Σ 1, 51, 2601, 132651, 6765201, …
  (−∞, −1]nonnegativerational roots
• A125092 → θ+1 → reverse
  1 · 8, 1 · 27, 16, 1 · 64, 81, 24, 1 … Σ 1, 9, 44, 170, 576, 1792, 5248, 14688, …
  (−∞, 0]nonnegative
• A142461 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 16, 16 · 1, 114, 336, 224 … Σ 1, 3, 33, 675, 19521, 732483, 33821793, …
  [−1, 0]nonnegative
• Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).
  32, -12, 1 · -384, 176, -24, 1 … Σ 21, -231, 3465, -65835, 1514205, …
  [2, ∞)rational roots
• A038305 → basis A013609 → basis A007318
  361, 228, 36 · 6859, 6498, 2052, 216 … Σ 625, 15625, 390625, 9765625, 244140625, …
  (−∞, −1]nonnegativerational roots
• A038320 → basis A013609 → basis A007318
  841, 696, 144 · 24389, 30276, 12528, 1728 … Σ 1681, 68921, 2825761, 115856201, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A297191, shifted → (θ+1)² → D
  0, 18 · 0, 0, 0, 100 · 0, 0, 0, 0, 0, 294 … Σ 0, 18, 100, 294, 648, 1210, 2028, 3150, …
  (−∞, 0]nonnegativerational roots
• A391533 → basis A013609 → D
  12, 8 · 42, 64, 24 · 120, 296, 240, 64 … Σ 20, 130, 720, 3680, 17920, 84480, …
  (−∞, 0]nonnegativerational roots
• A108838 → D → D+1
  2 · 12, 4 · 50, 42, 6 · 160, 264, 96, 8 … Σ 0, 2, 16, 98, 528, 2640, 12584, 58058, …
  (−∞, 0]nonnegative
• A132774 → D
  3 · 4, 10 · 0, 12, 21 · 0, 0, 24, 36 … Σ 0, 3, 14, 33, 60, 95, 138, 189, …
  (−∞, 0]nonnegativerational roots
• A130123 → D
  2 · 0, 8 · 0, 0, 24 · 0, 0, 0, 64 … Σ 0, 2, 8, 24, 64, 160, 384, 896, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729, shifted → basis A013609 → D+1
  1 · -3, -2 · 5, 12, 4 · -7, -30, -36, -8 … Σ 1, -5, 21, -81, 297, -1053, 3645, …
  (−∞, 0]rational roots
• A207606 → reverse
  2, 7, 4 · 2, 11, 16, 5 · 2, 15, 36, 30, 6 … Σ 13, 34, 89, 233, 610, 1597, 4181, …
  (−∞, 0]nonnegative
• A075501 → θ+1
  36, 36, 3 · 216, 504, 108, 4 … Σ 75, 832, 10721, 156264, 2525083, …
  (−∞, 0]nonnegative
• A249252 → basis A007318 → D+1
  1 · 9, 4 · 41, 40, 8 · 185, 284, 128, 16 … Σ 1, 13, 89, 613, 4081, 26621, 170889, …
  (−∞, 0]nonnegative
• A145596 → θ+1
  1 · 2, 4 · 3, 16, 9 · 4, 40, 60, 16 … Σ 1, 6, 28, 120, 495, 2002, 8008, 31824, …
  (−∞, 0]nonnegative
• A123519 → basis A007318 → (θ+1)²
  11, 56, 36 · 41, 304, 396, 128 … Σ 103, 869, 6877, 52079, 381827, 2730505, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → θ+1 → D+1
  1 · -2, -2 · 0, 6, 3 · 0, 0, -12, -4 … Σ 1, -4, 9, -16, 25, -36, 49, -64, …
  (−∞, 0]rational roots
• A327997 → x ↦ x+1 → θ+1
  1 · 4, 2 · 17, 18, 3 · 78, 130, 45, 4 … Σ 1, 6, 38, 257, 1873, 14802, 127278, …
  (−∞, 0]nonnegative
• A008459 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … Σ 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  [−1, 0]nonnegative
• A136124 → basis A013610 → reverse
  1 · 3, 3 · 9, 21, 12 · 27, 108, 141, 60 … Σ 1, 6, 42, 336, 3024, 30240, 332640, …
  (−∞, 0]nonnegativerational roots
• A257607 → D+1
  1 · 10, 5 · 85, 110, 25 · 660, 1605, 910, 125 … Σ 1, 15, 220, 3300, 51480, 840840, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/2) (shift +1) → x ↦ x+1 → x ↦ x+1
  1 · 1 · 3, 1 · 9, 4 … Σ 1, 1, 4, 13, 70, 385, 2710, 20665, …
  (−∞, −1]nonnegative
• A145324 → basis A007318 → x ↦ x/(x+1)
  1 · 3, 5 · 12, 41, 35 · 60, 313, 544, 315 … Σ 1, 8, 88, 1232, 20944, 418880, 9634240, …
  [−1, 0]nonnegativerational roots
• A130534 → D+1 → (θ+1)²
  1 · 2, 4 · 5, 20, 9 · 17, 92, 81, 16 … Σ 1, 6, 34, 206, 1388, 10466, 87974, …
  (−∞, 0]nonnegative
• A038326 → basis A013609 → D
  24 · 1104, 1152 · 38088, 79488, 41472 … Σ 0, 24, 2256, 159048, 9967008, …
  (−∞, 0]nonnegativerational roots
• A081578 → basis A013609 → D
  2 · 14, 8 · 60, 96, 24 · 244, 624, 408, 64 … Σ 0, 2, 22, 180, 1340, 9334, 62610, …
  (−∞, 0]nonnegative
• A144696 → (θ+1)²
  1 · 1, 8 · 1, 28, 36 · 1, 72, 297, 128 … Σ 1, 9, 65, 498, 4200, 39120, 400680, …
  (−∞, 0]nonnegative
• A357367 → θ+1
  1 · 0, 4 · 0, 12, 36 · 0, 48, 360, 480 … Σ 1, 4, 48, 888, 21960, 676800, 24927840, …
  (−∞, 0]nonnegative
• A196347 → basis A013609 → (θ+1)²
  1 · 2, 8 · 8, 64, 72 · 48, 576, 1296, 768 … Σ 1, 10, 144, 2688, 61440, 1658880, …
  (−∞, 0]nonnegative
• A094816 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 5, 20, 19 · 16, 94, 176, 106 … Σ 1, 6, 44, 392, 4176, 52192, 751552, …
  [−1, 0]nonnegative