Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A103633 → basis A007318 → x ↦ x+1
  1 · 2, 1 · 6, 5, 1 · 12, 16, 7, 1 … Σ 1, 3, 12, 36, 144, 432, 1728, 5184, …
  (−∞, −1]nonnegativerational roots
• A062190: Coefficient triangle of certain polynomials N(5; m,x).
  1 · 1, 6 · 1, 14, 21 · 1, 24, 84, 56 … Σ 1, 7, 36, 165, 715, 3003, 12376, 50388, …
  (−∞, 0)nonnegative
• A225471 → basis A008277 → x ↦ x+1
  1 · 5, 1 · 46, 15, 1 · 613, 258, 30, 1 … Σ 1, 6, 62, 902, 16878, 385582, 10394902, …
  (−∞, −1]nonnegative
• A241171 → D
  6 · 30, 180 · 126, 2520, 7560 … Σ 0, 6, 210, 10206, 707370, 67365606, …
  (−∞, 0]nonnegative
• A266213 → basis A013610 → x ↦ x+1
  1 · 1 · 9, 6 · 49, 60, 18 … Σ 1, 1, 15, 127, 1121, 9857, 86703, …
  (−∞, −1]nonnegative
• A243662 → x ↦ x/(x+1)
  1 · 3, 4 · 12, 32, 21 · 55, 220, 290, 126 … Σ 1, 7, 65, 691, 7953, 96447, 1213761, …
  [−1, 0]nonnegative
• A356546 → x ↦ x+1 → D
  2 · 24, 12 · 240, 240, 60 · 2240, 3360, 1680, 280 … Σ 0, 2, 36, 540, 7560, 102060, 1347192, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729 → D → (θ+1)²
  1 · 0, 8 · 0, 0, 27 · 0, 0, 0, 64 … Σ 0, 1, 8, 27, 64, 125, 216, 343, …
  (−∞, 0]nonnegativerational roots
• A013609 → basis A013610 → θ+1
  1 · 3, 12 · 9, 72, 108 · 27, 324, 972, 864 … Σ 1, 15, 189, 2187, 24057, 255879, …
  (−∞, 0]nonnegativerational roots
• A114687 → θ+1 → D+1
  1 · 5, 4 · 13, 36, 12 · 25, 168, 168, 32 … Σ 1, 9, 61, 393, 2481, 15473, 95677, …
  (−∞, 0]nonnegative
• A200536 → x ↦ x/(x+1)
  1 · 1, 5, 6 · 1, 10, 37, 60, 36 … Σ 1, 12, 144, 1728, 20736, 248832, …
  [−1, 0]nonnegativerational roots
• A127648 → (θ+1)² → θ+1
  1 · 0, 16 · 0, 0, 81 · 0, 0, 0, 256 … Σ 0, 1, 16, 81, 256, 625, 1296, 2401, …
  (−∞, 0]nonnegativerational roots
• A038323 → x ↦ x+1 → D+1
  1 · 29, 9 · 760, 522, 81 · 18800, 20520, 7047, 729 … Σ 1, 38, 1363, 47096, 1585285, 52338794, …
  (−∞, 0]nonnegativerational roots
• A225471 → basis A007318 → θ+1
  1 · 4, 2 · 32, 24, 3 · 384, 352, 72, 4 … Σ 1, 6, 59, 812, 14389, 312114, 8011695, …
  (−∞, 0]nonnegative
• A038305 → basis A013610 → D+1
  403, 396, 81 · 6760, 10881, 5346, 729 … Σ 880, 23716, 617584, 15695152, …
  (−∞, 0]nonnegativerational roots
• A105868 → x ↦ x+1
  1 · 1, 1 · 3, 4, 1 · 7, 15, 9, 1 … Σ 1, 2, 8, 32, 136, 592, 2624, 11776, …
  (−∞, −1]nonnegative
• A013612 → x ↦ x+1 → (θ+1)²
  36, 240, 225 · 216, 2160, 4050, 2000 … Σ 501, 8426, 130801, 1924626, 27246901, …
  (−∞, 0]nonnegative
• A142460 → x ↦ x+1
  1 · 2, 1 · 14, 14, 1 · 168, 252, 86, 1 … Σ 1, 3, 29, 507, 12461, 395643, 15414029, …
  (−∞, −1]nonnegative
• A028338 → x ↦ x+1 → D+1
  1 · 3, 1 · 14, 8, 1 · 92, 68, 15, 1 … Σ 1, 4, 23, 176, 1689, 19524, 264207, …
  (−∞, 0]nonnegative
• A125662 → basis A013610 → D
  3 · 24, 18 · 138, 216, 81 · 696, 1674, 1296, 324 … Σ 0, 3, 42, 435, 3990, 34260, 282198, …
  (−∞, 0]nonnegative
• shift_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
  1 · 4, 4, 1 · 24, 48, 34, 10, 1 … Σ 1, 9, 117, 2025, 43821, 1137969, …
  (−∞, −1]nonnegative
• A030528 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 24, 30, 9 · 116, 219, 135, 27 … Σ 1, 8, 63, 497, 3920, 30919, 243873, …
  (−∞, −1]nonnegative
• A122431 → D
  3, 6, 3 · 1, 6, 9, 4 · 0, 2, 9, 12, 5 … Σ 12, 20, 28, 36, 44, 52, 60, 68, …
  (−∞, 0]nonnegativerational roots
• A104684 → basis A007318 → (θ+1)²
  1 · 3, 4 · 13, 32, 9 · 63, 228, 135, 16 … Σ 1, 7, 54, 442, 3670, 30474, 252028, …
  (−∞, 0]nonnegative
• A225471 → basis A013610 → D+1
  1 · 7, 3 · 68, 54, 9 · 912, 960, 297, 27 … Σ 1, 10, 131, 2196, 45189, 1105182, …
  (−∞, 0]nonnegative
• A116414 → basis A013609 → θ+1
  1 · 5, 4 · 22, 40, 12 · 95, 276, 180, 32 … Σ 1, 9, 74, 583, 4461, 33416, 246289, …
  (−∞, 0]nonnegative
• A028326 → basis A013611 → x ↦ x/(x+1)
  2 · 4, 12 · 8, 48, 72 · 16, 144, 432, 432 … Σ 2, 16, 128, 1024, 8192, 65536, 524288, …
  [−1, 0]nonnegativerational roots
• Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).
  162, -27, 1 · -4374, 891, -54, 1 … Σ 136, -3536, 123760, -5445440, …
  [9/2, ∞)rational roots
• A075497 → reverse → x ↦ x+1
  1 · 3, 2 · 11, 14, 4 · 49, 92, 52, 8 … Σ 1, 5, 29, 201, 1657, 15821, 170389, …
  (−∞, −1]nonnegative
• A166961 → x ↦ x+1
  1 · 2, 1 · 11, 11, 1 · 103, 146, 45, 1 … Σ 1, 3, 23, 295, 5671, 149599, 5139791, …
  (−∞, −1]nonnegative
• A038323 → x ↦ x+1 → basis A007318
  1 · 29, 9 · 841, 522, 81 · 24389, 22707, 7047, 729 … Σ 1, 38, 1444, 54872, 2085136, 79235168, …
  (−∞, −1]nonnegativerational roots
• A039756 → reverse → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 16, 38, 24 … Σ 1, 3, 13, 79, 601, 5339, 53861, 607527, …
  [−1, 0]nonnegative
• A367270 → basis A013609 → D
  4 · 24, 24 · 120, 240, 96 · 560, 1680, 1440, 320 … Σ 0, 0, 4, 48, 456, 4000, 33720, 277536, …
  (−∞, 0]nonnegative
• A122542 → D → D
  2 · 8, 6 · 16, 36, 12 · 24, 108, 96, 20 … Σ 0, 0, 2, 14, 64, 248, 874, 2894, …
  (−∞, 0]nonnegative
• A249251 → basis A007318 → θ+1
  1 · 4, 6 · 13, 36, 18 · 43, 174, 171, 48 … Σ 1, 10, 67, 436, 2749, 16954, 102823, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A008288 → x ↦ x+1
  1 · 5, 5, 1 · 29, 58, 40, 11, 1 … Σ 1, 11, 139, 1763, 22363, 283667, …
  (−∞, −1]nonnegative
• A178822 → x ↦ x+1
  1 · 12, 6 · 84, 84, 21 · 448, 672, 336, 56 … Σ 1, 18, 189, 1512, 10206, 61236, 336798, …
  (−∞, −1]nonnegativerational roots
• A156366 → x ↦ x/(x+1)
  1, 14, 22 · 1, 36, 168, 160 · 1, 82, 834, 2128, 1456 … Σ 37, 365, 4501, 66605, 1149877, …
  [−1, 0]nonnegative
• A091042 → basis A013610 → θ+1
  1 · 4, 18 · 16, 120, 135 · 64, 672, 1512, 756 … Σ 1, 22, 271, 3004, 31981, 332146, …
  (−∞, 0]nonnegative
• A130123 → reverse
  1 · 2 · 4 · 8 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+2
• A094262 → x ↦ x+1
  1 · 4, 4, 1 · 32, 72, 60, 22, 3 … Σ 1, 9, 189, 6561, 318573, 19882017, …
  (−∞, −1]nonnegative
• A290319 → basis A013610 → θ+1
  1 · 2, 6 · 12, 48, 27 · 120, 552, 486, 108 … Σ 1, 8, 87, 1266, 23277, 518652, …
  (−∞, 0]nonnegative
• A116414 → basis A007318 → (θ+1)²
  1 · 5, 4 · 22, 40, 9 · 95, 276, 135, 16 … Σ 1, 9, 71, 522, 3663, 24867, 164673, …
  (−∞, 0]nonnegative
• A063967 → basis A013611 → reverse
  1 · 4, 2 · 16, 20, 6 · 64, 128, 80, 16 … Σ 1, 6, 42, 288, 1980, 13608, 93528, …
  (−∞, 0]nonnegative
• A176230 → θ+1
  1 · 1, 2 · 3, 12, 3 · 15, 90, 45, 4 … Σ 1, 3, 18, 154, 1692, 22596, 354328, …
  (−∞, 0]nonnegative
• A038231 → Veronese 2 → basis A007318
  1 · 4 · 17, 1 · 76, 12 … Σ 1, 4, 18, 88, 452, 2384, 12744, 68576, …
  (−∞, −1]nonnegative
• A038329 → basis A013609 → basis A007318
  441, 252, 36 · 9261, 7938, 2268, 216 … Σ 729, 19683, 531441, 14348907, …
  (−∞, −1]nonnegativerational roots
• A106246 → basis A007318
  1 · 3, 1 · 6, 6, 1 · 10, 18, 9, 1 … Σ 1, 4, 13, 38, 104, 272, 688, 1696, …
  (−∞, −1]nonnegative
• A111125 → reverse → x ↦ x/(x+1)
  1 · 1, 4 · 1, 7, 11 · 1, 10, 31, 29 … Σ 1, 5, 19, 71, 265, 989, 3691, 13775, …
  [−1, 0]nonnegative
• A225470 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 18, 63, 54 · 162, 783, 1242, 648 … Σ 1, 9, 135, 2835, 76545, 2525985, …
  [−1, 0]nonnegativerational roots