Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A106800 → θ+1
  1 · 1 · 1, 2 · 1, 6, 3 … Σ 1, 1, 3, 10, 38, 161, 747, 3753, …
  (−∞, 0]nonnegative
• A013612 → x ↦ x+1 → θ+1
  36, 120, 75 · 216, 1080, 1350, 500 … Σ 231, 3146, 41261, 527076, 6603091, …
  (−∞, 0]nonnegativerational roots
• A136572 → basis A013609
  1 · 1, 2 · 2, 8, 8 · 6, 36, 72, 48 … Σ 1, 3, 18, 162, 1944, 29160, 524880, …
  [−1, 0]nonnegativerational roots
• A081578 → basis A013609 → θ+1
  1 · 2, 4 · 7, 28, 12 · 20, 120, 144, 32 … Σ 1, 6, 47, 316, 2109, 13634, 86731, …
  (−∞, 0]nonnegative
• A038336 → x ↦ x/(x+1)
  144, 528, 484 · 1728, 9504, 17424, 10648 … Σ 1156, 39304, 1336336, 45435424, …
  [−1, 0]nonnegativerational roots
• A038259 → basis A013610 → θ+1
  1 · 11, 30 · 121, 660, 675 … Σ 1, 41, 1456, 47996, 1511536, 46154576, …
  (−∞, 0]nonnegativerational roots
• A038323 → basis A013610 → D+1
  1 · 47, 27 · 1480, 2538, 729 … Σ 1, 74, 4747, 282752, 16092565, …
  (−∞, 0]nonnegativerational roots
• A178820 → Veronese 2 → D
  10 · 60 · 210, 70 · 560, 560 … Σ 0, 0, 10, 60, 280, 1120, 4032, 13440, …
  (−∞, 0]nonnegative
• A013613 → basis A013611 → x ↦ x+1
  1 · 31, 24 · 961, 1488, 576 … Σ 1, 55, 3025, 166375, 9150625, …
  (−∞, −1]nonnegativerational roots
• A360282 → reverse
  1 · 1 · 2, 3 · 3, 12, 10 … Σ 1, 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, 0]nonnegative
• A207823 → reverse → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 24 · 1, 15, 73, 115 … Σ 1, 6, 35, 204, 1189, 6930, 40391, …
  [−1, 0]nonnegative
• A154986 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 10, 12 · 1, 25, 80, 60 … Σ 1, 4, 23, 166, 1437, 14488, 166515, …
  [−1, 0]nonnegative
• A144436 → D
  1 · 8, 2 · 23, 46, 3 · 54, 340, 162, 4 … Σ 0, 1, 10, 72, 560, 4680, 42672, 424480, …
  (−∞, 0]nonnegative
• A098158 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 0, 27, 16 … Σ 1, 4, 13, 43, 130, 366, 980, 2528, …
  (−∞, 0]nonnegative
• A028246 → D+1
  1 · 2, 1 · 4, 7, 2 · 8, 31, 30, 6 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−∞, 0]nonnegative
• A008955 → basis A007318 → basis A007318
  1 · 3, 1 · 27, 21, 4 · 513, 642, 265, 36 … Σ 1, 4, 52, 1456, 71344, 5422144, …
  (−∞, −1]nonnegativerational roots
• A008292 → γ → x ↦ x+1
  1 · 1 · 3, 2 · 9, 8 … Σ 1, 1, 5, 17, 109, 649, 5285, 44513, …
  (−∞, −1]nonnegative
• A001263 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … Σ 1, 5, 22, 95, 406, 1722, 7260, 30459, …
  (−∞, 0]nonnegative
• A028338 → D
  1 · 4, 2 · 23, 18, 3 · 176, 172, 48, 4 … Σ 0, 1, 6, 44, 400, 4384, 56448, 836352, …
  (−∞, 0]nonnegative
• chebyshev_t → D+1
  1 · 1, 1 · -1, 4, 2 · -3, -3, 12, 4 … Σ 1, 2, 5, 10, 17, 26, 37, 50, …
  (−∞, ∞)
• A147716 → basis A013611 → reverse
  1 · 4, 15 · 16, 120, 225 · 64, 720, 2700, 3375 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  (−∞, 0]nonnegativerational roots
• A135065 → D+1
  1 · 8, 4 · 27, 36, 9 · 64, 144, 96, 16 … Σ 1, 12, 72, 320, 1200, 4032, 12544, …
  (−∞, −1]nonnegativerational roots
• A397048 → reverse
  1 · 1, 1 · 1, 5, 4 · 1, 12, 37, 27 … Σ 1, 2, 10, 77, 799, 10427, 163967, …
  (−∞, 0]nonnegative
• A091042 → basis A007318 → D+1
  1 · 7, 3 · 36, 30, 5 · 176, 224, 77, 7 … Σ 1, 10, 71, 484, 3229, 21198, 137411, …
  (−∞, 0]nonnegative
• A139548 → basis A013609 → D
  16, 16 · 256, 768, 768, 256 … Σ 0, 32, 2048, 98304, 4194304, 167772160, …
  (−∞, 0]nonnegativerational roots
• A086645 → D+1
  1 · 2, 1 · 7, 8, 1 · 16, 45, 18, 1 … Σ 1, 3, 16, 80, 384, 1792, 8192, 36864, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 (shift +1) → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 10, 23, 15 … Σ 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  [−1, 0]nonnegative
• A099097 → x ↦ x−15
  1 · -45, 3 · 2010, -269, 9 · -89775, 18045, -1209, 27 … Σ 1, -42, 1750, -72912, 3037804, …
  (131/9, 15]
• A257625: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 6*…
  1 · 3, 3 · 9, 54, 9 · 27, 621, 621, 27 … Σ 1, 6, 72, 1296, 31104, 933120, …
  (−∞, 0)nonnegative
• A144697 → (θ+1)²
  1 · 1, 12 · 1, 40, 81 · 1, 100, 603, 432 … Σ 1, 13, 122, 1136, 11200, 118920, …
  (−∞, 0]nonnegative
• A113214 → D
  1 · 3, 2 · 2, 8, 3 · 0, 10, 15, 4 … Σ 0, 1, 5, 13, 29, 60, 118, 225, …
  (−∞, 0]nonnegative
• A110441 → reverse → x ↦ x+1
  1 · 4, 3 · 14, 20, 7 · 48, 100, 68, 15 … Σ 1, 7, 41, 231, 1289, 7175, 39913, …
  (−∞, −1]nonnegative
• A052553 → basis A013611 → basis A007318
  1 · 1 · 6, 4 · 11, 8 … Σ 1, 1, 10, 19, 109, 280, 1261, 3781, …
  (−∞, −1]nonnegative
• A132393 → θ+1 → D
  2 · 2, 6 · 4, 18, 12 · 12, 66, 72, 20 … Σ 0, 2, 8, 34, 170, 998, 6776, 52400, …
  (−∞, 0]nonnegative
• A085880 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 8, 32, 18 · 40, 240, 270, 80 … Σ 1, 6, 58, 630, 7182, 83916, 994356, …
  (−∞, 0]nonnegative
• A013622 → basis A013610 → θ+1
  1 · 8, 30 · 64, 480, 675 · 512, 5760, 16200, 13500 … Σ 1, 38, 1219, 35972, 1009861, 27424418, …
  (−∞, 0]nonnegativerational roots
• A003506 → (θ+1)² → (θ+1)²
  1 · 2, 32 · 3, 96, 243 · 4, 192, 972, 1024 … Σ 0, 1, 34, 342, 2192, 11000, 47232, …
  (−∞, 0]nonnegative
• A196347 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 8, 40, 50 · 48, 360, 900, 750 … Σ 1, 7, 98, 2058, 57624, 2016840, …
  [−1, 0]nonnegativerational roots
• A119808 → (θ+1)²
  1 · 2, 4 · 4, 20 · 8, 68, 18 … Σ 1, 6, 24, 94, 356, 1314, 4752, 16902, …
  (−∞, 0]nonnegative
• A111577 → (θ+1)²
  1 · 1, 4 · 1, 20, 9 · 1, 84, 108, 16 … Σ 1, 5, 30, 209, 1663, 14856, 146695, …
  (−∞, 0]nonnegative
• A143494 → x ↦ x+1 → D
  1 · 7, 2 · 40, 24, 3 · 221, 206, 54, 4 … Σ 0, 1, 9, 67, 485, 3571, 27181, 215195, …
  (−∞, 0]nonnegative
• A106246 → D+1 → x ↦ x/(x+1)
  1 · 3, 4 · 5, 16, 12 · 3, 24, 48, 28 … Σ 1, 7, 33, 103, 241, 471, 817, 1303, …
  [−1, 0]nonnegative
• A130123 → θ+1 → reverse
  1 · 4 · 12 · 32 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0]nonnegativerational roots
• A059364 → basis A013609 → reverse
  1 · 2, 3 · 8, 22, 15 · 48, 188, 244, 105 … Σ 1, 5, 45, 585, 9945, 208845, 5221125, …
  (−∞, 0]nonnegativerational roots
• A103633 → basis A007318 → D+1
  1 · 2, 1 · 5, 5, 1 · 7, 13, 7, 1 … Σ 1, 3, 11, 28, 96, 228, 756, 1728, …
  (−∞, 0]nonnegative
• A274496 → (θ+1)²
  1 · 2 · 2, 8 · 4, 16 … Σ 1, 2, 10, 20, 72, 144, 448, 896, …
  (−∞, 0]nonnegative
• A105278 → reverse
  1 · 1, 2 · 1, 6, 6 · 1, 12, 36, 24 … Σ 1, 3, 13, 73, 501, 4051, 37633, 394353, …
  (−∞, 0]nonnegative
• A395972 → D
  16, 16 · 72, 288, 144 · 256, 2304, 4608, 1536 … Σ 32, 504, 8704, 167200, 3561984, …
  (−∞, 0]nonnegative
• A038220 → basis A013610 → x ↦ x/(x+1)
  1 · 5, 11 · 25, 110, 121 · 125, 825, 1815, 1331 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  [−1, 0]nonnegativerational roots
• A106246 → θ+1 → D+1
  1 · 4, 2 · 9, 14, 3 · 6, 42, 30, 4 … Σ 1, 6, 26, 82, 207, 446, 856, 1506, …
  (−∞, 0]nonnegative